Rahmati, A., Nejati Barzoki, F. (2018). Using Burnett Equations to Derive an Analytical Solution to PressureDriven Gas Flow and Heat Transfer in MicroCouette Flow. Journal of Heat and Mass Transfer Research(JHMTR), 5(2), 8794. doi: 10.22075/jhmtr.2018.3658
Ahmad Reza Rahmati; Faezeh Nejati Barzoki. "Using Burnett Equations to Derive an Analytical Solution to PressureDriven Gas Flow and Heat Transfer in MicroCouette Flow". Journal of Heat and Mass Transfer Research(JHMTR), 5, 2, 2018, 8794. doi: 10.22075/jhmtr.2018.3658
Rahmati, A., Nejati Barzoki, F. (2018). 'Using Burnett Equations to Derive an Analytical Solution to PressureDriven Gas Flow and Heat Transfer in MicroCouette Flow', Journal of Heat and Mass Transfer Research(JHMTR), 5(2), pp. 8794. doi: 10.22075/jhmtr.2018.3658
Rahmati, A., Nejati Barzoki, F. Using Burnett Equations to Derive an Analytical Solution to PressureDriven Gas Flow and Heat Transfer in MicroCouette Flow. Journal of Heat and Mass Transfer Research(JHMTR), 2018; 5(2): 8794. doi: 10.22075/jhmtr.2018.3658
Using Burnett Equations to Derive an Analytical Solution to PressureDriven Gas Flow and Heat Transfer in MicroCouette Flow
^{}Department of Mechanical Engineering, University of Khashan, Kashan, Iran
Receive Date: 29 December 2016,
Revise Date: 05 October 2017,
Accept Date: 12 October 2017
Abstract
The aim of the present study is deriving an analytical solution to incompressible thermal flow in a microCouette geometry in the presence of a pressure gradient using Burnett equations with first and secondorder slip boundary conditions. The lower plate of the microCouette structure is stationary, whereas the upper plate moves at a constant velocity. Nondimensional axial velocity and temperature profiles were obtained using the slip boundary conditions and were compared with the transition flow regime (0.1≤ Kn ≤10). The results showed that in this regime, rarefaction exerts a considerable effect on both the velocity and the temperature profiles. Because of the presence of the pressure gradient in the direction of the flow, the nondimensional velocity and temperature profiles behave in a parabolic trend and flatten as the Knudsen number increases. The Poiseuille and Nusselt numbers, obtained using the derived analytical solution, decrease with increasing the Knudsen number. In the absence of an axial pressure gradient, the velocity profile behaves linearly and shows good agreement with the results of the previous works.
In recent years, the study of gas flow and heat transfer in microscale devices has become more prevalent in many scientific fields, e.g., microelectromechanical systems and the potential applications of microdevices in engineering, medical sciences. Such interest is largely driven by the need for new technical tools which accurately predict the microscale transport processes and devices that are characterized by performanceenhancing designs [1]. Recent advances in micro/nanotechnology have enabled the manufacture of mechanical devices such as microchannels, nozzles, and pumps. With these devices, porous media can be employed for microfiltration, fractionation, catalysis, and microbiologyrelated applications. Microdevices have also benefitted from the use of charged porous media structures, which are intended to magnify pumping, mixing, and separation effects [2].
Explorations into gas flow and its heat transfer characteristics in microgeometries typically take the form of numerical and experimental research (e.g., [3–5]). In such studies, microCouette flow is regarded as an important problem in fluid mechanics since it is widely implicated in hydrodynamic lubrication, polymer/food processing, and viscometry, among other processes [6]. It also provides insight into the generation of unsteady boundary layers. A Couette flowbased approach is a suitable modeling method for microflows and a simple analogue for applications such as harddisc drive reader heads, microturbines, and gas bearings. For the purpose of examining Couette
H
x
y
U_{0}
P_{i}
P_{o}
flow, researchers perform a Chapman–Enskog expansion of the Boltzmann equation with the Knudsen number as a parameter, subsequently producing zeroorder Euler equations, firstorder Navier–Stokes equations, and secondorder Burnett equations [7, 8]. Xue et al. [9], for instance, analyzed microCouette flow and heat transfer using Burnett equations. The authors found that rarefaction exerts a significant effect on the distribution of velocity, temperature, and pressure when flow enters the transition regime in isothermal walls or when the temperature ratio is high in nonisothermal walls.
Walls and Abedian [10] investigated the bivelocity gas dynamics of microchannel Coquette flow to elucidate the characteristics of monoatomic Maxwellian gas ﬂow in the microchannel. Their results indicated that numerical predictions of density, temperature, and velocity distributions differ from those obtained using Burnett equations. Singh et al. [11] analytically solved the Burnett equations for plane Poiseuille flow and obtained a normalized mass flow rate, friction factor, and axial velocity profile which reveals a very good agreement with their experimental and simulation data. The analytical solution derived by the authors also predicts changes in the curvature of a streamwise pressure profile. Singh et al. [12] analyzed a solution of plane Couette ﬂow in the transition regime and compared it with direct simulation Monte Carlo (DSMC) data. Good agreement between the results was observed up to a Knudsen number (Kn) equal to 10. Their results also enabled the formulation of a slip relationship, which can potentially provide a more accurate slip velocity in the transition regime than those from Maxwell’s slip model. Lockerby and Reese [13] conducted highresolution Burnett simulations of microCouette ﬂow and heat transfer. The results on highresolution numerical grids showed good agreement with data obtained from direct simulation methods. Xue et al. [14] analyzed the gaseous microCouette ﬂow utilizing the DSMC method, Navier–Stokes equations, and Burnette equations. Finally, they compared the results of the three approaches.
Several researchers have also investigated the effects of pressuredriven flow in microchannels. Zahid et al. [15], for instance, examined the Couette–Poiseuille ﬂow of a gas in long microchannels. In their work, the gas was shear driven and subjected to a pressure gradient. They comprehensively studied the effects of rarefaction and compressibility on ﬂow characteristics and found the parallel ﬂow assumption to be invalid in cases characterized
Figure 1. Couette flow coordinate system.
by slight rarefaction. Finally, the authors found that axial and vertical velocity components depend on the degree of rarefaction, utilized pressure gradient, and wall velocity. Beskok et al. [16] carried out a detailed examination of the effects of rarefaction and compressibility on pressuredriven and sheardriven microﬂows. The authors discovered that rarefaction and compressibility phenomena need to be considered in microﬂuidic investigation. Jang and Wereley [17] studied the pressure distribution of gaseous ﬂow in rectangular microchannels and revealed that the deviation of pressure distributions from linear distribution decreases with augmentation of rarefaction. Bao and Lin [18] performed a Burnett simulation of compressible gas flow and heat transfer in microPoiseuille flow in the slip and transition ﬂow regimes. It was concluded that with enhancement of Knudsen number, the Poiseuill number decreases and the Nusselt number would increase.
The present study extends the mentioned earlier works by using Burnett equations in order to capture an analytical solution to microCouette flow in the presence of a pressure gradient. The research concentrated on the flow and heat transfer behaviors of the Couette flow and compared these with first and secondorder boundary conditions. The results manifested that in the presence of a pressure gradient, the velocity profile converts from a linear profile into a parabolic one. Furthermore, an increase in the Knudsen number elevates the slip on the wall of the examined geometry and smoothens the temperature and velocity profiles.
Burnett equations applied to microCouette ﬂow
In the present work, gas flow between two parallel infinite flat plates was considered the Couette flow (Fig. 1). The space between the plates is separated by a distance H. The lower plate is stationary, whereas the upper plate moves at a constant velocity U0. The gas ﬂow is driven by the pressure difference between inlet pressure pi and outlet pressure po. The Burnett equations were simplified under the assumption that the flow is steady, twodimensional, fully developed, and incompressible. A pressure gradient was applied in the axial direction.
The general tensor expression of Burnettlevel stress tensor [19] is presented in Eq. (1).
(1)
Where coefficients depend on the gas model. The Couette length (L) is considered to be much greater than the Couette height (H). Thus,
Burnett equations can be extensively simplified, thus generating terms that include up to . The results of this expansion, including those on terms, are presented in [19]. With terms disregarded, x and y momentum equations can be substantially simplified as follows:
(2)
(3)
Under the assumption of a Maxwellian gas model, for which coefficients = (10/3, 2, 8) [20], and nondimensionalization with reference exit conditions , the following equation would be obtained:
(4)
Similarly, for the y momentum equation, the following expression could be derived:
(5)
Where and refer to the Knudsen and Mach numbers estimated at the outlet of the studied geometry respectively. These numbers are expressed as follows:
(6)
(7)
The term is relatively small for low Mach number flows in the early transition regime. In this case, for flow through a very long channel, the Burnett equations are reduced to:
(8)
(9)
Slip boundary conditions
Nonslip boundary conditions are generally unrealistic for transition flows since the collisions near a wall are insufficient to equilibrate the flow field. Slip conditions should therefore be considered at such wall. Some researchers solved Burnett equations with firstorder slip boundary conditions [9, 13], while others used the equations with secondorder slip boundary conditions [11, 19]. The present work deviates from convention by imposing both types of boundary conditions on the calculations.
3.1. Firstorder slip boundary conditions
The firstorder slip boundary conditions are written as follows [13]:
(10)
(11)
where and are the tangential momentum and thermal accommodation coefficients, respectively, and subscripts s and w denote slip and wall values, respectively. In these conditions, the effects of thermal creep and quadratic variation with are disregarded.
After nondimensionalization with characteristic length L, Eqs. (10) and (11) take the forms
(12)
(13)
where (∂/∂n*) indicates gradients normal to the wall surface, and = n/L.
3.2. Secondorder slip boundary conditions
Some researchers believe that because Burnett equations are secondorder solutions of the Boltzmann equation, the boundary conditions also require to be accurate at the secondorder level in relation to Kn. Assuming that = 1 and , the general secondorder slip conditions have the following nondimensional forms:
(14)
(15)
Where coefficients A_{1} and A_{2} are the slip coefficients. Different typical values of slip coefficients were expanded by researchers [21]. In the present study,the values in the Deissler model [22] are adopted, which indicates that A_{1} = 1 and A_{2} =–9/8. In Eq. (15), Pr and are the Prandtl number and the ratio of specific heat, respectively.
Solution procedure and results
On the basis of Eq. (8), velocity distribution can be modeled as a parabolic phenomenon in the transition regime, with a compatible slip condition as follows:
(16)
Where reflects the functional dependence of velocity on pressure gradient, viscosity, channel height, and local mean free path.
The integration of the velocity distribution derived from Eq. (14) over the crosssection of the microCouette geometry leads to the mean velocity as below:
(17)
The utilization of the firstorder slip boundary conditions in Eq. (12) for the two upper and lower walls of the microCouette geometry renders the problem in the following form:
(18)
(19)
It was assumed that . Finally, solving Eq. (16) yields the following coefficients:
(20)
(21)
Accordingly, the normalized velocity of flow in the microCouette geometry is reduced to:
(22)
Eq. (16) is solved again, this time with the secondorder slip boundary conditions [22], and only coefficient b is modified as follows:
(23)
The normalized velocity can also be changed into:
(24)
Heat transfer is an important issue in Couette flow. Thus, Burnett equations are provided as the sum of corresponding Navier–Stokes and Burnett heat flux terms:
= (25)
For the Navier–Stokes equations, the terms are as follows:
(26)
Where k is the thermal conductivity of gas, and T denotes the temperature. For Burnett equations, for a fully developed incompressible flow in the microCouette geometry is the same as that in Navier–Stokes equations. A microCouette geometry with a lower adiabatic wall ( ) and an upper wall with constant heat flux ( ) was considered. Flow was supposed to be hydrodynamically and thermally developed and incompressible, and fluid properties were assumed to be uniform. The energy equation is reduced into:
(27)
Disregarding viscous dissipation, the following parameter was introduced:
(28)
Where , and is the convection heat transfer coefficient. Since the flow is thermally developed, the energy equation could be simplified as:
(29)
The conservation of energy on a control volume of length dx and height H yields , thereby Eq. (29) would be reduced to:
(30)
(31)
With thermal creep disregarded, the firstorder slip boundary conditions are:
(32)
(33)
Using a monatomic ideal gas with Pr = 2/3 and = 5/3, it is possible to reduce the Eq. (33) to:
(34)
The integration of Eq. (31) with boundary conditions (32) and (34) yields:
(35)
The coefficient of m can be obtained as:
Eq. (31) is also solved with the secondorder slip boundary conditions. Thus,
(37)
(38)
In this case, the coefficient of m can be modified as follows:
(39)
Discussion
In microCouette geometry where the transition ﬂow regime is subjected to a pressure gradient of zero, Burnett equations are reduced to:
(40)
In order to validate the results, gas flow was considered as the ﬂow between two infinitely long and wide parallel plates separated by a distance H. These plates move in opposite directions with equal velocity (U_{O}/2). The integration of Eq. (40) with the firstorder slip boundary conditions therefore generates a linear velocity profile as:
(41)
The values obtained from the equation above for Kn = 0.3, 0.5 show good agreement with those obtained by Singh et al. [12] via the DSMC method (Fig. 2). The small error at Kn = 0.5 is due to the accuracy of the method by which the Burnett equations were simplified.
Figs. 3 and 4 present the normalized velocity profile obtained from the Burnett equations with first and secondorder boundary conditions, respectively. The normalized velocity obtained on the basis of the firstorder boundary conditions (Eq. (22)) approach and the normalized velocity derived with the secondorder boundary conditions (Eq. (24)). The velocity profile flattens with increasing Knudsen number. Consequently, it could be concluded that with increasing the Knudsen number, the slip on the wall improves, and the effect of upper plate velocity would decrease.
Figure 2. Comparison of nondimensional velocity proﬁles in the present work and in Singh et al. [12] (via DSMC) for different Kn numbers.
Figure 3. Variations in normalized velocity across the channel, derived using Burnett equations with firstorder slip boundary conditions.
Figure 4. Variations in normalized velocity across the channel, derived using Burnett equations with secondorder slip boundary conditions.
Figure 5. Nondimensional temperature profile derived using Burnett equations with firstorder slip boundary conditions.
Figure 6. Nondimensional temperature profile derived using Burnett equations with secondorder slip boundary conditions.
Figure 7. Variations in the product of the friction factor and Reynolds number with Knudsen number.
Figure 8. Variations in Nu with Kn.
Another integral parameter of interest to the engineering group is the skin friction coefficient ( ). The Poiseuille number (Po) is explained as the product of the friction factor and Reynolds number as:
(42)
(43)
(44)
Fig. 7, which presents variations in the Poiseuille number with Knudsen number, reveals that the former decreases with an increase in the latter.
With Eq. (28), T and are calculated, and the Nusselt number is obtained as:
(45)
Fig. 8 demonstrates that Nu decreases with increase of the Kn. This behavior can be described on the basis of Fig. 4., which indicates that with increasing Knudsen number, rapidly increases, thereby reducing Nu (see Eq. (45)).
The present study was an attempt to the analytical solution of pressure and sheardriven microCouette ﬂow in terms of Burnett equations in the transition regime. The equations were simplified and solved for stress and heat flux terms by imposing first and secondorder slip boundary conditions on microCouette flow. In the absence of an axial pressure gradient, the velocity profile behaves linearly, which indicates a good agreement with the results available in the literature. By applying a pressure gradient in the direction of flow, the velocity profile behaved parabolically.
The normalized velocity obtained via the firstorder slip boundary conditions is approximately similar to that derived through the secondorder slip boundary conditions. However, the nondimensional temperature profiles obtained using the conditions considerably differ at Kn>1. Additionally, the nondimensional velocity and temperature profiles at a large Kn are ﬂatter than those at a small Kn. This finding is due to the fact that an increasing Knudsen number elevates the slip on the wall and decreases the curvature of the profiles. Furthermore, the Poiseuille and Nusselt numbers rapidly decrease with increase of the Kn.
To sum up, the Burnett equations have been manifested to be an extremely good technique for solution of simple gas ﬂows in microﬂuidic applications, especially for high values of the Kn number. As for a future work, Burnett equations could be utilized for more complex twodimensional geometries and threedimensional micro flow problems.
Nomenclature
C_{f}
friction factor
C_{p}
heat capacity
H
microCouette height
k
thermal conductivity
Kn
Knudsen number
L
microCouette length
M
Mach number
Nu
Nusselt number
Pr
Prandtl number
p
pressure
q
heat flux
R
speciﬁc gas constant
Re
Reynolds number
T
temperature
u
axial velocity
v
vertical velocity
Greek symbols
α
convection heat transfer coefﬁcient
ratio of specific heat
ratio of channel height to length
mean free path
viscosity
density
thermal accommodation coefficient
tangential momentum accommodation coefficient
shear stress
Superscripts and subscripts
i
inlet
o
outlet
s
slip
x,y
crosssectional coordinates
w
wall
∗
nondimensional parameters
**
normalized velocity

averaged quantities
[.1]The authors can strengthen the conclusion section by providing a discussion of the implications of your results and/or recommendations for future research.
Answer: it is modified
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