Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis
M. E. Sayed-Ahmed1*, A. Saif-Elyazal1and L. Iskander2
1Department of Engineering Mathematics and Physics, Faculty of Engineering, Fayoum University, Fayoum-63111, Egypt
2Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza-12163, Egypt
Abstract
Laminar fully developed flow and heat transfer of Herschel-Bulkley fluids through rectangular duct is investigated numerically. The non-linear momentum and energy equations are solved numerically using finite-element approximations. We consider two cases of thermal boundary conditions H1and T thermal boundary conditions. The velocity, temperature profiles, product of friction factor-Reynolds number and Nusselt number for H1and T thermal boundary conditions are computed for various values of the physical parameters of the Herschel-Bulkley fluids and aspect ratio of the duct. The present results have been compared with the known solution for Newtonian and power-law fluids and are found to be in good agreement.
Key Words: Numerical Analysis, Non-Newtonian Fluids, Heat Transfer, Rectangular Duct
1. Introduction
The fluid flow behavior of non-Newtonian fluid has attracted special interest in recent years due to the wide application of these fluids in the chemical, pharmaceu- tical, petrochemical, food industries and electronic in- dustries. A large number of fluids that are used exten- sively in industrial application are non-Newtonian fluids exhibiting a yield stressty, stress that has to be exceeded before the fluid moves. As a result the fluid cannot sus- tain a velocity gradient unless the magnitude of the local shear stress is higher than this yield stress. Fluids that belong to this category include cement, drilling mud, sludge, granular suspensions, aqueores foams, slurries, paints, plastics, paper pulp and food products.
Several studies have appeared, [1-6] studied the fully developed velocity profile and fraction factor- Reynolds number product and Nusselt number for New- tonian in a rectangular duct. In early studies for non- Newtonian fluid, Schechter [7] applied a variational
method to obtain the velocity profile and the corre- sponding friction factor for a power-law fluid. Wheeler and Wissler [8] solved the same problem by finite differ- ence method and furthermore proposed a simple friction factor-Reynolds number correlation for the special case of a square duct. Kozicki et al. [9] presented fairly approximate relationship between friction factor and Reynolds number through rectangular ducts as well as for some other non-circular ducts. More recent studies in the field under discussion are mainly based on numerical approaches. Geo and Hartnett [10] used finite difference method to obtain friction factor-Reynolds number pro- duct and velocity profile for power-law fluid through rectangular duct. On the other hand Seppo Serjala [11, 12] solved the same problem by using finite-element method. The flow of Herschel-Bulkley fluids has been considered for other geometries by Bettra and Eissa [13]
in entrance region through parallel plate channels. Eissa [14] studied laminar heat transfer for thermally deve- loping flow of Herschel-Bulkley fluid through special cases of square ducts by using finite difference method.
Sayed-Ahmed and El-yazal [15,16] studied the flow and
*Corresponding author. E-mail: [email protected]
heat transfer of Robertson-Stiff fluids in rectangular duct using finite-difference method.
In the present study, we investigate the problem of laminar fully developed flow of Herschel-Bulkley fluids through rectangular duct. The non-linear momentum equation is solved iteratively using a finite-element met- hod to obtain the velocity profile and the value of the friction factor-Reynolds number product. The energy equation is solved numerically to obtain the temperature profile and Nusselt number for two cases of thermal boundary conditions H1 (axially uniform heat flux and peripherally uniform temperature) and T (axially and peripherally uniform temperature). Computations are given over wide range of duct aspect ratios and flow behavior index and yield stress of Herschel-Bulkley fluids.
2. Problem Formulation
We consider a steady, fully developed, laminar, iso- thermal flow of in-compressible and purely viscous non-Newtonian in a rectangular duct. The duct con- figuration and coordinate system are shown in Figure 1.
For the previous assumptions, the momentum and energy equations reduced to
(1)
(2)
In which w is the velocity component in z-direction and the apparent viscosityh for Herschel-Bulkley fluids is described by
(3)
The boundary condition for the velocity is no slip boundary condition (i.e. the velocity at the walls of a rectangular duct equals zero). The boundary conditions for the temperature is T = Tw (the temperature at the
wall). Since the flow is hydro dynamically and thermally fully developed flow then the term¶T/¶z in the energy equation (2) can be represented as [12]
(4)
for case (1): H1thermal boundary condition, and as
(5)
for case (2): T thermal boundary condition
The fraction factor f and Reynolds number Re for Herschel-Bulkley fluids are given by
(6)
The average velocity wavis given by
(7)
Now it is convenient to write the above equations in the non-dimensional form. The relevant dimensionless qu- antities are defined by
(8)
Substitution these quantities in equation (8) into
Figure 1. Configuration and coordinate system for a rectan- gular duct.
equations (1) and (3), and by making use of the relation (6) and (7). The dimensionless equation of motion is reduced to
(9)
and the dimensionless viscositym is reduced to
(10)
The dimensionless energy equations are reduced to for case (1): H1thermal boundary condition
(11)
and case (2): T thermal boundary condition
(12)
Noting that for symmetry reasons only a quarter of the flow domain has to be considered (0£ X £ 0.5, 0 £ Y £ 0.5a), the velocity boundary conditions can be written as
(13)
and the temperature boundary conditions can be written as
(14)
The product of friction factor and Reynolds number reduce into the dimensionless form
(15)
where dimensionless average velocity Wavcan be writ- ten as
(16)
The Nusselt number Nu is defined by
(17)
where h is the average heat transfer coefficient and given by
(18)
The Nusselt number for the two cases thermal boun- dary condition is given by
(19)
whereqmis the bulk mean temperature and evaluated by the form
(20)
3. Numerical Solution
The finite-element method is used to approximate the solution of partial differential equations (9), (11) and (12) with boundary conditions (13) and (14). We will di- vided the domain into finite rectangular elements, which is a suitable element for the domain as shown in Figure 2 with sides DX and DY. Suppose that W and q can be approximated by the expression
(21)
whereyjis a linear interpolation function of the rectan- gular element.
Multiply equation (9), (11) and (12) by a test func- tion v and integrate over the element domain We. The variational form becomes
(22)
(23)
(24) Substituting equation (21) for W,q and yifor v into the variational form (22)-(24), we obtain
(25)
(26)
(27)
wherem is the average viscosity m over the element and q is the average dimensionless temperature q over the element from the last iteration.
The linear interpolation functions for rectangular duct with both sides’DX and DY are given by [17]
(28)
Using linear interpolation functions in equations (28) into equations (25), (26) and (27) and evaluate the integrals then substituting the boundary conditions to get a system of linear equations on Wjandqj. For non linear equation (25) an iterative procedure is used to obtain the unknown Wj. The initial value of viscosity is assumed to be the same at each mesh node (m = 1.0 for Newtonian fluid) to obtain the velocity at each node of the elements.
Making use the value of the velocity that are obtain to get the new viscosity by
(29)
The process is repeated until the criteria of conver- gence W(i, j)new -W(i, j)old £10-5 and mnew(i, j) -m(i, j)old £10-5 are satisfied. The convergence has been achieved by taking 40´ 40 elements. The average velocity Wmis evaluated by Simpson rule formula of the double integrals
(30)
For case (1): H1thermal boundary condition, equation (26) is solved to obtain the temperatureqj. For case (2):
T thermal boundary conditions, non linear equation (27) an iterative procedure is used to obtain the un- knownqj. We assume initial value of temperature the Figure 2. The domain is divided into rectangular elements
with four nodes.
same at each mesh node (q = 1.0) and evaluating the bulk mean temperature to obtain the temperature at each node of the elements. Making use the new value of the temperature that are obtain to get the new value of the bulk mean temperature. The process is repeated un- til the criteria of convergence q(i, j) q
new (i, j)
- old £10-5 are satisfied. The convergence has been achieved by taking 40´ 40 elements.
4. Results and Discussions
The velocity profile and friction factor fRe are ob- tained for different values of (n = 1.5, 1.2, 1.0, 0.8, 0.5), (tD = 0.0, 0.01, 0.03) and (a = 1.0, 0.5, 0.2). Figures
3a-3c, 4a-4c and 5a-5c show the distribution of velo- city W along the centerline of the cross-section, which is parallel to the major side of the duct, for various values of n,tDand fora = 1.0, 0.5, 0.2 respectively.
The study of Figures (3-5) shows that the value of the velocity W increases from zero at the wall (no slip condition) to the maximum value at the mid point (0.5, 0.5a) for all values of a, n and tD. The maximum value of the velocity increases as the n increases for all val- ues oftDanda as a result of decrease in the apparent viscosity of the non-Newtonian fluid and, therefore, increasing the average velocity. Also the value of the velocity decreases with increasing intDfor all values of n anda due to increasing the plug core formation,
Figure 3. The distribution of velocity W along the centerline of the cross-section, which is parallel to the major side of the duct fora = 1.0.
Figure 4. The distribution of velocity W along the centerline of the cross-section, which is parallel to the major side of the duct fora = 0.5.
which no flow occurs through the region, so the net flow region decreasing. The velocity gradient at the wall increases as n increases for all values oftDanda.
We can also observe that the velocity profile becomes flatter with a decrease in the value of the flow behavior index n (shear thinning n < 1.0) anda due to decreas- ing in apparent viscosity of the non-Newtonian fluid.
It is found also the velocity profile becomes increas- ingly flattened with increasing of the value oftDthis phenomenon is due to the formulation of the plug core of the yield non-Newtonian fluids along the centerline of the duct.
The variation of fRe with flow behavior index n is shown in Figures (6a-6c) for a = 1.0, 0.5, 0.2 respec-
tively. It has been found that the value of fRe increases with increasing value of n due to the increasing of the viscosity of Herschel-Bulkley fluid with the flow in- dex n for all values oftDanda. It can found also the increasing of the value oftDincreases the value of fRe for all values of n and a. This phenomenon is ex- plained as follows an increasing in the value of the yield stresstDincreases the plug core formation, which no flow occurs through this region and net flow region decreasing. The decreasing in the value ofa increases the value of fRe for all values of n andtDas a result of decreasing the cross section area.
Nusselt number are obtained for different values of (n = 1.5, 1.2, 1.0, 0.8, 0.5), (tD= 0.0, 0.01, 0.03) and (a = 1.0, 0.5, 0.2). Figures 7a-7c and 8a-8c show the varia- tion of Nusselt number for case (1): H1thermal boundary Figure 5. The distribution of velocity W along the centerline
of the cross-section, which is parallel to the major side of the duct fora = 0.2.
Figure 6. The variation of fRe with flow behavior index n.
conditions and case (2): T thermal boundary conditions for various values of n,tD and fora = 1.0, 0.5, 0.2 re- spectively. Examination of Figures 7a-7c and 8a-8c shows that the value of Nu for case (1) and case (2) ther- mal boundary conditions increases as the flow behavior index n decreases for all values oftDanda. The increase of tD increases Nu for case (1) and case (2) thermal boundary conditions for all values of n anda. This phe- nomenon is explained as follows an increasing in the value of the yield stresstD increases the plug core for- mation, which no flow occurs through this region and net flow region decreasing which decreases bulk mean temperature. The decreasing in the value ofa increases the value of Nu for case (1) and case (2) thermal boun- dary conditions for all values of n andtDas a result of
decreasing the cross section area, which decreases bulk mean temperature. It also found that, the values of Nu for H1thermal boundary conditions greater than the values of Nu for T thermal boundary conditions.
Tables 1-3 show the comparison of the present re- sults of fRe and Nu for H1 and T thermal boundary condition with the previous work (Shah and London [5] and Syrjala [11,12]) fora = 1.0,0.5,0.2, n = 1.2, 1.0, 0.5, andtD= 0.0 (power-law fluid). The present results are found to be in good agreement with the previous work.
Nomenclature
Dh hydraulic diameter (4 cross area/perimeter) Figure 8. The variation of Nusselt number Nu with flow be-
havior index n for case (2): T thermal boundary condition.
Figure 7. The variation of Nusselt number Nu with flow be- havior index n for case (1): H1thermal boundary condition.
F fraction factor
fi(e) element load force vector h average heat transfer coefficient
k thermal conductivity of Herschel-Bulkley fluids kij(e) element stiffness matrix
L maximum length of the rectangular duct m consistency index of Herschel-Bulkley fluids n flow behaviour index of Herschel-Bulkley fluids Nu Nusselt number
P Pressure
qn secondary variable projection along unit vector Re Reynolds number
T temperature
Tm bulk mean temperature Tw wall temperature v test function
w axial fluid velocity in duct wav average fluid velocity in duct W dimensionless axial velocity in duct Wav dimensionless average fluid velocity x, y, z rectangular Cartesian coordinates X, Y, Z dimensionless coordinate in x, y, z axes a aspect ratio
k thermal diffusivity
h apparent viscosity of the model m dimensionless viscosity of the model
m average dimensionless viscosity over the ele- ment
q dimensionless temperature qm the bulk mean temperature
q is the average dimensionless temperatureq over the element
ty yield stress value of the model tD dimensionless yield stress We The element domain
Ge The boundary of the element yj linear interpolation function
DX, DY sides length of the rectangular element References
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n 1.0 0.5 1.0 0.5 1.0 0.5
Shah and London, [5] 14.227 15.548 19.071
Syrjala, [11] 14.227 5.721 15.548 5.999 19.071 6.800
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Manuscript Received: Nov. 29, 2006 Accepted: Sep. 5, 2007
