Hindawi Publishing Corporation International Journal of Chemical Engineering Volume 2013, Article ID 310273, 4 pages http://dx.doi.org/10.1155/2013/310273 Research Article Flow Reversal of Fully Developed Mixed Convection in a Vertical Channel with Chemical Reaction Habibis Saleh, 1 Ishak Hashim, 1,2 and Sri Basriati 3 1 School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia 2 Solar Energy Research Institute, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia 3 Mathematics Sciences, UIN Sultan Syarif Kasim Riau, Panam, Pekanbaru 28293, Indonesia Correspondence should be addressed to Habibis Saleh; [email protected] Received 12 March 2013; Accepted 21 April 2013 Academic Editor: Donald L. Feke Copyright © 2013 Habibis Saleh et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e present analysis is concerned with the criteria for the onset of flow reversal of the fully developed mixed convection in a vertical channel under the effect of the chemical reaction. e governing equations and the critical values of the buoyancy force are solved and calculated numerically via MAPLE. Parameter zones for the occurrence of reversed flow are presented. e exothermic chemical reaction is found to enhance the flow reversal and made flow reversal possible for symmetrical walls temperature. 1. Introduction e study of combined and free convection flow in a heated vertical parallel-plate channel has received considerable attention because of its wide range of applications such as geothermal reservoir, cooling of nuclear reactors, thermal insulation, energy storage, and conservation, chemical, food, and metallurgical industries, and petroleum reservoirs. A combined forced and natural convective flow puts on the situation where an externally driven flow joins with an native buoyant flow and when both are prominent. One of the earliest studies on mixed convection in a vertical channel with uniform wall temperatures had been studied analytically by Tao [1]. en Habchi and Acharya [2] studied mixed con- vection in a vertical channel with asymmetric walls heating, where one plate is heated and the other is adiabatic. Fully developed flow was revisited by Aung and Worku [3] and Cheng et al. [4]. ey analyzed effects of various boundary conditions and different constant temperature parameters on buoyancy-aided flow. Recently, Pop et al. [5] studied the mixed convection in a vertical channel with chemical reaction and found that dual solutions exist for both velocity and temperature. e high buoyancy force due to a differentially heated wall combined with an upward flow leads to a high fluid flow adjacent to the walls that can precipitate a downward flow (i.e., reversal flow) emanating from the open top of the channel in order to augment the increased upward flow. Experiment on the appearance of flow reversal was conducted by Sparrow et al. [6] for free convection in a one-sided heated vertical channel. Aung and Worku [3] and Cheng et al. [4] studied analytically the occurrence flow reversal for the mixed convection. Criteria for the onset of flow reversal in terms of the Grashof to Reynolds number ratio were developed. Criteria for the occurrence of this flow, adjacent to the colder wall under effects of concentration, and magnetic field including internal heating, micropolar as well as thermophoretic, were presented by [7–9] and [10, 11], respectively. Numerical studies by Pop et al. [5] found that reversed flow also occurs near the hot wall, where physically it is not possible [3]. ey demonstrated that increasing the internal heating and constant temperature parameters could enhance the reversed flow but overlooked determining the criteria for the onset of flow reversal. erefore, in this paper, we shall present the conditions for the onset of flow reversal of the fully developed flow mixed convection flow in a vertical channel under effect of the chemical reaction for various values of the constant temperature parameters studied by Pop et al. [5].
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Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2013 Article ID 310273 4 pageshttpdxdoiorg1011552013310273
Research ArticleFlow Reversal of Fully Developed Mixed Convection ina Vertical Channel with Chemical Reaction
Habibis Saleh1 Ishak Hashim12 and Sri Basriati3
1 School of Mathematical Sciences Universiti Kebangsaan Malaysia 43600 Bangi Selangor Malaysia2 Solar Energy Research Institute Universiti Kebangsaan Malaysia 43600 Bangi Selangor Malaysia3Mathematics Sciences UIN Sultan Syarif Kasim Riau Panam Pekanbaru 28293 Indonesia
Correspondence should be addressed to Habibis Saleh drhabibissalehgmailcom
Received 12 March 2013 Accepted 21 April 2013
Academic Editor Donald L Feke
Copyright copy 2013 Habibis Saleh et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The present analysis is concerned with the criteria for the onset of flow reversal of the fully developed mixed convection in avertical channel under the effect of the chemical reactionThe governing equations and the critical values of the buoyancy force aresolved and calculated numerically via MAPLE Parameter zones for the occurrence of reversed flow are presented The exothermicchemical reaction is found to enhance the flow reversal and made flow reversal possible for symmetrical walls temperature
1 Introduction
The study of combined and free convection flow in a heatedvertical parallel-plate channel has received considerableattention because of its wide range of applications such asgeothermal reservoir cooling of nuclear reactors thermalinsulation energy storage and conservation chemical foodand metallurgical industries and petroleum reservoirs Acombined forced and natural convective flow puts on thesituation where an externally driven flow joins with an nativebuoyant flow and when both are prominent One of theearliest studies onmixed convection in a vertical channelwithuniform wall temperatures had been studied analytically byTao [1] Then Habchi and Acharya [2] studied mixed con-vection in a vertical channel with asymmetric walls heatingwhere one plate is heated and the other is adiabatic Fullydeveloped flow was revisited by Aung and Worku [3] andCheng et al [4] They analyzed effects of various boundaryconditions and different constant temperature parameterson buoyancy-aided flow Recently Pop et al [5] studiedthe mixed convection in a vertical channel with chemicalreaction and found that dual solutions exist for both velocityand temperature
The high buoyancy force due to a differentially heatedwall combined with an upward flow leads to a high fluid
flow adjacent to the walls that can precipitate a downwardflow (ie reversal flow) emanating from the open top ofthe channel in order to augment the increased upward flowExperiment on the appearance of flow reversal was conductedby Sparrow et al [6] for free convection in a one-sidedheated vertical channel Aung and Worku [3] and Chenget al [4] studied analytically the occurrence flow reversalfor the mixed convection Criteria for the onset of flowreversal in terms of the Grashof to Reynolds number ratiowere developed Criteria for the occurrence of this flowadjacent to the colder wall under effects of concentrationand magnetic field including internal heating micropolar aswell as thermophoretic were presented by [7ndash9] and [10 11]respectively
Numerical studies by Pop et al [5] found that reversedflow also occurs near the hot wall where physically it is notpossible [3] They demonstrated that increasing the internalheating and constant temperature parameters could enhancethe reversed flow but overlooked determining the criteriafor the onset of flow reversal Therefore in this paper weshall present the conditions for the onset of flow reversal ofthe fully developed flow mixed convection flow in a verticalchannel under effect of the chemical reaction for variousvalues of the constant temperature parameters studied by Popet al [5]
2 International Journal of Chemical Engineering
2 Mathematical Formulation
Consider the steady flow of a viscous and incompressiblefluid between two vertical and parallel plane walls Thedistance between the walls that is the channel width is ℓ Acoordinate system is chosen such that the 119909-axis is parallel tothe gravitational acceleration vector g but with the oppositedirection The 119910-axis is orthogonal to the channel walls andthe origin of the axes is such that the positions of the channelwalls are 119910 = 0 and 119910 = ℓ respectively A sketch of the systemand of the coordinate axes is reported in Figure 1The walls at119910 = 0 and 119910 = ℓ are isothermal at given temperatures 119879
1and
1198792 where we assume that 119879
1ge 1198792 The fluid has a uniform
vertical upward stream wise velocity distribution 1198800at the
channel entrance Thus the basic equations for steady fullydeveloped flow of a viscous incompressible fluid and the heatassumed supplied to the surrounding fluid by an exothermicsurface reaction [5] are
119892120573 (119879 minus 1198790) minus
1
120588
119889119901
119889119909+ ]
1198892119906
1198891199102= 0
1205721198892119879
1198891199102+ 1198761198960119886119890minus119864119877119879
= 0
(1)
subject to the boundary conditions
119906 (0) = 0 119906 (ℓ) = 0
119879 (0) = 1198791 119879 (ℓ) = 119879
2 (119879
1ge 1198792)
(2)
Reaction takes place only on the surface which is governedby the single first-order Arrhenius kinetics The closuresystem by mass flux conservation 119872 is int119871
0119880119889119884 = 119872 The
dimensionless governing equations for the problem are [5]
1198892119880
1198891198842= minus119866119877Θ + 119860 (3)
1198892Θ
1198891198842= minus119870119865119890Θ (4)
subject to the boundary conditions
119880 (0) = 0 119880 (1) = 0
Θ (0) = 119877119879 Θ (1) = minus119877
119879
(5)
And the global conservation of mass at any cross section inthe channel is int1
0119880119889119884 = 1
3 Solutions Method
Equations (3) and (4) subject to (2) have been solved nu-merically by Pop et al [5] using the bvpp4c function fromMATLAB Pop et al [5] found the solutions are possible onlyfor the finite range of 119870
119865 The range depend on constant
temperature parameter 119877119879 for example at 119877
119879= 1 the
solution domain are in 0 le 119870119865lt 3155 We solved again (3)
and (4) subject to (5) via dsolve fromMAPLEThe nonlinear-ities of (4) were handled by applying continuation optionalin dsolve The continuation is utilized to maintain Newtonmethod (inside package dsolve) reaching the convergence
119909
Chemical reaction
1198791 1198792
119910 = 0
1198800
119910 = 985747
119910
g
Figure 1 Schematic representation of the model
4 Results and Discussion
The effects of various levels 119866119877 119870119865 and 119877
119879are presented
in Figure 2 These figures show that velocity increases as119866119877 119870119865 or 119877
119879increase Flow reversal is found near the
colder wall (119884 = 1) for 119866119877
ge 50 Increasing the Frank-Kamenetskii number or constant temperature parameterenhances the flow reversal The interesting behavior of thereversed flowoccurs at119877
119879= 01 in Figure 2(c) where reversal
is found near both of the walls This phenomenon due toheat generated by exothermic surface reaction and beingconducted away into the surrounding fluid and this behaviorprogress continuously cause the rate of the increase in thetemperature to rise sharply in middle region of the channel
The result in Figure 2 also implies that there exists acritical value119866
119877119888such that for119866
119877gt 119866119877119888 flow reversal occurs
at 119884 = 1 By setting 119889119880119889119884 = 0 at 119884 = 1 and utilizing dsolvewith continuation the119866
119877119888can be obtainedwithin the solution
domain The numerical value 119866119877119888
is plotted in Figure 3It was observed that increasing the Frank-Kamenetskii
number reduces 119866119877119888
(Figure 3(a)) while reducing the tem-perature difference ratio increases the 119866
119877119888(Figure 3(b))
When119870119865= 0 we recovered the critical value which had been
calculated by Aung and Worku [3] Cheng et al [4] El-Din[7] Barletta [12] and Chamkha et al [9] We can not obtainthe critical value of the ratio of Grashof number andReynoldsnumber (119866
119877) with symmetrical wall temperatures 119877
119879= 0
when there is no chemical reaction in the fluidThe119866119877119888exists
only in the range of constant 119877119879le 005
5 Conclusions
Parameter zones for the occurrences flow reversal by mixedconvection under the effect of chemical reaction in a verticalparallel-plate channel are presented We can conclude thatflow reversal adjacent to the cold wall is found to exist withinthe channel as the ratio of Grashof number and Reynoldsnumber is above a threshold value The exothermic chemical
International Journal of Chemical Engineering 3
119866119877 = 300
119866119877 = 50
119866119877 = 10
119910
119906
1080604020
6
4
2
0
minus2
minus4
(a)
119910
119906
1080604020
6
4
2
0
minus2
minus6
minus4
119870119865 = 31
119870119865 = 15
119870119865 = 0
(b)
119910
119906
1080604020
6
4
2
0
minus2
minus6
minus4
119877119879 = 01
119877119879 = 05
119877119879 = 1
(c)
Figure 2 Plots of 119880 versus 119884 for different values of (a) 119866119877where 119870
119865= 15 119877
119879= 1 (b) 119870
119865where 119866
119877= 300 119877
119879= 1 and (c) 119877
119879where
119866119877= 300 119870
119865= 31
3252151050
36
34
32
30
28
26
24
22
119870119865
119866119877119888
(a)
800
700
600
500
400
300
200
100
01 08 06 04 02 0
119877119879
119870119865 = 31
119870119865 = 15
119870119865 = 0
119866119877119888
(b)
Figure 3 The critical value 119866119877119888for the onset of a flow reversal with (a) increasing 119870
119865and (b) reducing 119877
119879
4 International Journal of Chemical Engineering
reaction is found to enhance the flow reversal and made flowreversal possible for symmetrical walls temperatures Theaspect of the nonuniqueness of the solution as found by Popet al [5] will be the focus of our next research undertaking
Nomenclature
119886 Concentration of reactant119860 Axial pressure gradient 119889119901119889119909Gr Grashof number119866119877 Dimensionless parameter GrRe
119892 Gravitational acceleration119864 Activation energy1198960 Preexponential factor
119870119865 Frank-Kamenetskii number
ℓ Channel width119876 Exothermicity factor119877 Universal gas constantRe Reynolds number119877119879 Temperature difference ratio
119879 Temperature119880 Dimensionless velocity components in
the 119909-direction119883 119884 Dimensionless space coordinates
Greek Symbols
120572 Thermal diffusivity120573 Thermal expansion coefficient120588 DensityΘ Dimensionless temperature120583 Dynamic viscosity] Kinematic viscosity
Subscript
0 Reference value1 Left wall2 Right wall119888 Critical
References
[1] L N Tao ldquoOn combined free and forced convection inchannelsrdquo Journal of Heat Transfer vol 82 pp 233ndash238 1960
[2] S Habchi and S Acharya ldquoLaminar mixed convection ina symmetrically or asymmetrically heated vertical channelrdquoNumerical Heat Transfer A vol 9 no 5 pp 605ndash618 1986
[3] W Aung and G Worku ldquoTheory of fully developed combinedconvection including flow reversalrdquo Journal of Heat Transfervol 108 pp 299ndash304 1986
[4] C H Cheng H S Kou and W H Huang ldquoFlow reversal andheat transfer of fully developed mixed convection in verticalchannelsrdquo Journal of Thermophysics and Heat Transfer vol 4no 3 pp 375ndash383 1990
[5] I Pop T Grosan and R Cornelia ldquoEffect of heat generated byan exothermic reaction on the fully developed mixed convec-tion flow in a vertical channelrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 471ndash4742010
[6] E M Sparrow G M Chrysler and L F Azevedo ldquoObservedflow reversals and measuredpredicted nusselt numbers fornatural convection in a one-sided heated vertical channelrdquoJournal of Heat Transfer vol 106 no 2 pp 325ndash332 1984
[7] M M S El-Din ldquoFully developed forced convection in avertical channel with combined buoyancy forcesrdquo InternationalCommunications in Heat and Mass Transfer vol 19 no 2 pp239ndash248 1992
[8] A J Chamkha ldquoOn laminar hydromagnetic mixed convectionflow in a vertical channel with symmetric and asymmetric wallheating conditionsrdquo International Journal of Heat and MassTransfer vol 45 no 12 pp 2509ndash2525 2002
[9] A J Chamkha T Grosan and I Pop ldquoFully developed mixedconvection of a micropolar fluid in a vertical channelrdquo Inter-national Journal of Fluid Mechanics Research vol 30 no 3 pp251ndash263 2003
[10] T Grosan R Pop and I Pop ldquoThermophoretic deposition ofparticles in fully developedmixed convection flow in a parallel-plate vertical channelrdquo Heat and Mass Transfer vol 45 no 4pp 503ndash509 2009
[11] E Magyari ldquoThermophoretic deposition of particles in fullydeveloped mixed convection flow in a parallel-plate verticalchannel the full analytical solutionrdquo Heat and Mass Transfervol 45 no 11 pp 1473ndash1482 2009
[12] A Barletta ldquoLaminar mixed convection with viscous dissipa-tion in a vertical channelrdquo International Journal of Heat andMass Transfer vol 41 no 22 pp 3501ndash3513 1998
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mechanical Engineering
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Advances inAcoustics ampVibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspJournalthinspofthinsp
Sensors
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Antennas andPropagation
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
2 International Journal of Chemical Engineering
2 Mathematical Formulation
Consider the steady flow of a viscous and incompressiblefluid between two vertical and parallel plane walls Thedistance between the walls that is the channel width is ℓ Acoordinate system is chosen such that the 119909-axis is parallel tothe gravitational acceleration vector g but with the oppositedirection The 119910-axis is orthogonal to the channel walls andthe origin of the axes is such that the positions of the channelwalls are 119910 = 0 and 119910 = ℓ respectively A sketch of the systemand of the coordinate axes is reported in Figure 1The walls at119910 = 0 and 119910 = ℓ are isothermal at given temperatures 119879
1and
1198792 where we assume that 119879
1ge 1198792 The fluid has a uniform
vertical upward stream wise velocity distribution 1198800at the
channel entrance Thus the basic equations for steady fullydeveloped flow of a viscous incompressible fluid and the heatassumed supplied to the surrounding fluid by an exothermicsurface reaction [5] are
119892120573 (119879 minus 1198790) minus
1
120588
119889119901
119889119909+ ]
1198892119906
1198891199102= 0
1205721198892119879
1198891199102+ 1198761198960119886119890minus119864119877119879
= 0
(1)
subject to the boundary conditions
119906 (0) = 0 119906 (ℓ) = 0
119879 (0) = 1198791 119879 (ℓ) = 119879
2 (119879
1ge 1198792)
(2)
Reaction takes place only on the surface which is governedby the single first-order Arrhenius kinetics The closuresystem by mass flux conservation 119872 is int119871
0119880119889119884 = 119872 The
dimensionless governing equations for the problem are [5]
1198892119880
1198891198842= minus119866119877Θ + 119860 (3)
1198892Θ
1198891198842= minus119870119865119890Θ (4)
subject to the boundary conditions
119880 (0) = 0 119880 (1) = 0
Θ (0) = 119877119879 Θ (1) = minus119877
119879
(5)
And the global conservation of mass at any cross section inthe channel is int1
0119880119889119884 = 1
3 Solutions Method
Equations (3) and (4) subject to (2) have been solved nu-merically by Pop et al [5] using the bvpp4c function fromMATLAB Pop et al [5] found the solutions are possible onlyfor the finite range of 119870
119865 The range depend on constant
temperature parameter 119877119879 for example at 119877
119879= 1 the
solution domain are in 0 le 119870119865lt 3155 We solved again (3)
and (4) subject to (5) via dsolve fromMAPLEThe nonlinear-ities of (4) were handled by applying continuation optionalin dsolve The continuation is utilized to maintain Newtonmethod (inside package dsolve) reaching the convergence
119909
Chemical reaction
1198791 1198792
119910 = 0
1198800
119910 = 985747
119910
g
Figure 1 Schematic representation of the model
4 Results and Discussion
The effects of various levels 119866119877 119870119865 and 119877
119879are presented
in Figure 2 These figures show that velocity increases as119866119877 119870119865 or 119877
119879increase Flow reversal is found near the
colder wall (119884 = 1) for 119866119877
ge 50 Increasing the Frank-Kamenetskii number or constant temperature parameterenhances the flow reversal The interesting behavior of thereversed flowoccurs at119877
119879= 01 in Figure 2(c) where reversal
is found near both of the walls This phenomenon due toheat generated by exothermic surface reaction and beingconducted away into the surrounding fluid and this behaviorprogress continuously cause the rate of the increase in thetemperature to rise sharply in middle region of the channel
The result in Figure 2 also implies that there exists acritical value119866
119877119888such that for119866
119877gt 119866119877119888 flow reversal occurs
at 119884 = 1 By setting 119889119880119889119884 = 0 at 119884 = 1 and utilizing dsolvewith continuation the119866
119877119888can be obtainedwithin the solution
domain The numerical value 119866119877119888
is plotted in Figure 3It was observed that increasing the Frank-Kamenetskii
number reduces 119866119877119888
(Figure 3(a)) while reducing the tem-perature difference ratio increases the 119866
119877119888(Figure 3(b))
When119870119865= 0 we recovered the critical value which had been
calculated by Aung and Worku [3] Cheng et al [4] El-Din[7] Barletta [12] and Chamkha et al [9] We can not obtainthe critical value of the ratio of Grashof number andReynoldsnumber (119866
119877) with symmetrical wall temperatures 119877
119879= 0
when there is no chemical reaction in the fluidThe119866119877119888exists
only in the range of constant 119877119879le 005
5 Conclusions
Parameter zones for the occurrences flow reversal by mixedconvection under the effect of chemical reaction in a verticalparallel-plate channel are presented We can conclude thatflow reversal adjacent to the cold wall is found to exist withinthe channel as the ratio of Grashof number and Reynoldsnumber is above a threshold value The exothermic chemical
International Journal of Chemical Engineering 3
119866119877 = 300
119866119877 = 50
119866119877 = 10
119910
119906
1080604020
6
4
2
0
minus2
minus4
(a)
119910
119906
1080604020
6
4
2
0
minus2
minus6
minus4
119870119865 = 31
119870119865 = 15
119870119865 = 0
(b)
119910
119906
1080604020
6
4
2
0
minus2
minus6
minus4
119877119879 = 01
119877119879 = 05
119877119879 = 1
(c)
Figure 2 Plots of 119880 versus 119884 for different values of (a) 119866119877where 119870
119865= 15 119877
119879= 1 (b) 119870
119865where 119866
119877= 300 119877
119879= 1 and (c) 119877
119879where
119866119877= 300 119870
119865= 31
3252151050
36
34
32
30
28
26
24
22
119870119865
119866119877119888
(a)
800
700
600
500
400
300
200
100
01 08 06 04 02 0
119877119879
119870119865 = 31
119870119865 = 15
119870119865 = 0
119866119877119888
(b)
Figure 3 The critical value 119866119877119888for the onset of a flow reversal with (a) increasing 119870
119865and (b) reducing 119877
119879
4 International Journal of Chemical Engineering
reaction is found to enhance the flow reversal and made flowreversal possible for symmetrical walls temperatures Theaspect of the nonuniqueness of the solution as found by Popet al [5] will be the focus of our next research undertaking
Nomenclature
119886 Concentration of reactant119860 Axial pressure gradient 119889119901119889119909Gr Grashof number119866119877 Dimensionless parameter GrRe
119892 Gravitational acceleration119864 Activation energy1198960 Preexponential factor
119870119865 Frank-Kamenetskii number
ℓ Channel width119876 Exothermicity factor119877 Universal gas constantRe Reynolds number119877119879 Temperature difference ratio
119879 Temperature119880 Dimensionless velocity components in
the 119909-direction119883 119884 Dimensionless space coordinates
Greek Symbols
120572 Thermal diffusivity120573 Thermal expansion coefficient120588 DensityΘ Dimensionless temperature120583 Dynamic viscosity] Kinematic viscosity
Subscript
0 Reference value1 Left wall2 Right wall119888 Critical
References
[1] L N Tao ldquoOn combined free and forced convection inchannelsrdquo Journal of Heat Transfer vol 82 pp 233ndash238 1960
[2] S Habchi and S Acharya ldquoLaminar mixed convection ina symmetrically or asymmetrically heated vertical channelrdquoNumerical Heat Transfer A vol 9 no 5 pp 605ndash618 1986
[3] W Aung and G Worku ldquoTheory of fully developed combinedconvection including flow reversalrdquo Journal of Heat Transfervol 108 pp 299ndash304 1986
[4] C H Cheng H S Kou and W H Huang ldquoFlow reversal andheat transfer of fully developed mixed convection in verticalchannelsrdquo Journal of Thermophysics and Heat Transfer vol 4no 3 pp 375ndash383 1990
[5] I Pop T Grosan and R Cornelia ldquoEffect of heat generated byan exothermic reaction on the fully developed mixed convec-tion flow in a vertical channelrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 471ndash4742010
[6] E M Sparrow G M Chrysler and L F Azevedo ldquoObservedflow reversals and measuredpredicted nusselt numbers fornatural convection in a one-sided heated vertical channelrdquoJournal of Heat Transfer vol 106 no 2 pp 325ndash332 1984
[7] M M S El-Din ldquoFully developed forced convection in avertical channel with combined buoyancy forcesrdquo InternationalCommunications in Heat and Mass Transfer vol 19 no 2 pp239ndash248 1992
[8] A J Chamkha ldquoOn laminar hydromagnetic mixed convectionflow in a vertical channel with symmetric and asymmetric wallheating conditionsrdquo International Journal of Heat and MassTransfer vol 45 no 12 pp 2509ndash2525 2002
[9] A J Chamkha T Grosan and I Pop ldquoFully developed mixedconvection of a micropolar fluid in a vertical channelrdquo Inter-national Journal of Fluid Mechanics Research vol 30 no 3 pp251ndash263 2003
[10] T Grosan R Pop and I Pop ldquoThermophoretic deposition ofparticles in fully developedmixed convection flow in a parallel-plate vertical channelrdquo Heat and Mass Transfer vol 45 no 4pp 503ndash509 2009
[11] E Magyari ldquoThermophoretic deposition of particles in fullydeveloped mixed convection flow in a parallel-plate verticalchannel the full analytical solutionrdquo Heat and Mass Transfervol 45 no 11 pp 1473ndash1482 2009
[12] A Barletta ldquoLaminar mixed convection with viscous dissipa-tion in a vertical channelrdquo International Journal of Heat andMass Transfer vol 41 no 22 pp 3501ndash3513 1998
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mechanical Engineering
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Advances inAcoustics ampVibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspJournalthinspofthinsp
Sensors
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Antennas andPropagation
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
International Journal of Chemical Engineering 3
119866119877 = 300
119866119877 = 50
119866119877 = 10
119910
119906
1080604020
6
4
2
0
minus2
minus4
(a)
119910
119906
1080604020
6
4
2
0
minus2
minus6
minus4
119870119865 = 31
119870119865 = 15
119870119865 = 0
(b)
119910
119906
1080604020
6
4
2
0
minus2
minus6
minus4
119877119879 = 01
119877119879 = 05
119877119879 = 1
(c)
Figure 2 Plots of 119880 versus 119884 for different values of (a) 119866119877where 119870
119865= 15 119877
119879= 1 (b) 119870
119865where 119866
119877= 300 119877
119879= 1 and (c) 119877
119879where
119866119877= 300 119870
119865= 31
3252151050
36
34
32
30
28
26
24
22
119870119865
119866119877119888
(a)
800
700
600
500
400
300
200
100
01 08 06 04 02 0
119877119879
119870119865 = 31
119870119865 = 15
119870119865 = 0
119866119877119888
(b)
Figure 3 The critical value 119866119877119888for the onset of a flow reversal with (a) increasing 119870
119865and (b) reducing 119877
119879
4 International Journal of Chemical Engineering
reaction is found to enhance the flow reversal and made flowreversal possible for symmetrical walls temperatures Theaspect of the nonuniqueness of the solution as found by Popet al [5] will be the focus of our next research undertaking
Nomenclature
119886 Concentration of reactant119860 Axial pressure gradient 119889119901119889119909Gr Grashof number119866119877 Dimensionless parameter GrRe
119892 Gravitational acceleration119864 Activation energy1198960 Preexponential factor
119870119865 Frank-Kamenetskii number
ℓ Channel width119876 Exothermicity factor119877 Universal gas constantRe Reynolds number119877119879 Temperature difference ratio
119879 Temperature119880 Dimensionless velocity components in
the 119909-direction119883 119884 Dimensionless space coordinates
Greek Symbols
120572 Thermal diffusivity120573 Thermal expansion coefficient120588 DensityΘ Dimensionless temperature120583 Dynamic viscosity] Kinematic viscosity
Subscript
0 Reference value1 Left wall2 Right wall119888 Critical
References
[1] L N Tao ldquoOn combined free and forced convection inchannelsrdquo Journal of Heat Transfer vol 82 pp 233ndash238 1960
[2] S Habchi and S Acharya ldquoLaminar mixed convection ina symmetrically or asymmetrically heated vertical channelrdquoNumerical Heat Transfer A vol 9 no 5 pp 605ndash618 1986
[3] W Aung and G Worku ldquoTheory of fully developed combinedconvection including flow reversalrdquo Journal of Heat Transfervol 108 pp 299ndash304 1986
[4] C H Cheng H S Kou and W H Huang ldquoFlow reversal andheat transfer of fully developed mixed convection in verticalchannelsrdquo Journal of Thermophysics and Heat Transfer vol 4no 3 pp 375ndash383 1990
[5] I Pop T Grosan and R Cornelia ldquoEffect of heat generated byan exothermic reaction on the fully developed mixed convec-tion flow in a vertical channelrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 471ndash4742010
[6] E M Sparrow G M Chrysler and L F Azevedo ldquoObservedflow reversals and measuredpredicted nusselt numbers fornatural convection in a one-sided heated vertical channelrdquoJournal of Heat Transfer vol 106 no 2 pp 325ndash332 1984
[7] M M S El-Din ldquoFully developed forced convection in avertical channel with combined buoyancy forcesrdquo InternationalCommunications in Heat and Mass Transfer vol 19 no 2 pp239ndash248 1992
[8] A J Chamkha ldquoOn laminar hydromagnetic mixed convectionflow in a vertical channel with symmetric and asymmetric wallheating conditionsrdquo International Journal of Heat and MassTransfer vol 45 no 12 pp 2509ndash2525 2002
[9] A J Chamkha T Grosan and I Pop ldquoFully developed mixedconvection of a micropolar fluid in a vertical channelrdquo Inter-national Journal of Fluid Mechanics Research vol 30 no 3 pp251ndash263 2003
[10] T Grosan R Pop and I Pop ldquoThermophoretic deposition ofparticles in fully developedmixed convection flow in a parallel-plate vertical channelrdquo Heat and Mass Transfer vol 45 no 4pp 503ndash509 2009
[11] E Magyari ldquoThermophoretic deposition of particles in fullydeveloped mixed convection flow in a parallel-plate verticalchannel the full analytical solutionrdquo Heat and Mass Transfervol 45 no 11 pp 1473ndash1482 2009
[12] A Barletta ldquoLaminar mixed convection with viscous dissipa-tion in a vertical channelrdquo International Journal of Heat andMass Transfer vol 41 no 22 pp 3501ndash3513 1998
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mechanical Engineering
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Advances inAcoustics ampVibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspJournalthinspofthinsp
Sensors
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Antennas andPropagation
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
4 International Journal of Chemical Engineering
reaction is found to enhance the flow reversal and made flowreversal possible for symmetrical walls temperatures Theaspect of the nonuniqueness of the solution as found by Popet al [5] will be the focus of our next research undertaking
Nomenclature
119886 Concentration of reactant119860 Axial pressure gradient 119889119901119889119909Gr Grashof number119866119877 Dimensionless parameter GrRe
119892 Gravitational acceleration119864 Activation energy1198960 Preexponential factor
119870119865 Frank-Kamenetskii number
ℓ Channel width119876 Exothermicity factor119877 Universal gas constantRe Reynolds number119877119879 Temperature difference ratio
119879 Temperature119880 Dimensionless velocity components in
the 119909-direction119883 119884 Dimensionless space coordinates
Greek Symbols
120572 Thermal diffusivity120573 Thermal expansion coefficient120588 DensityΘ Dimensionless temperature120583 Dynamic viscosity] Kinematic viscosity
Subscript
0 Reference value1 Left wall2 Right wall119888 Critical
References
[1] L N Tao ldquoOn combined free and forced convection inchannelsrdquo Journal of Heat Transfer vol 82 pp 233ndash238 1960
[2] S Habchi and S Acharya ldquoLaminar mixed convection ina symmetrically or asymmetrically heated vertical channelrdquoNumerical Heat Transfer A vol 9 no 5 pp 605ndash618 1986
[3] W Aung and G Worku ldquoTheory of fully developed combinedconvection including flow reversalrdquo Journal of Heat Transfervol 108 pp 299ndash304 1986
[4] C H Cheng H S Kou and W H Huang ldquoFlow reversal andheat transfer of fully developed mixed convection in verticalchannelsrdquo Journal of Thermophysics and Heat Transfer vol 4no 3 pp 375ndash383 1990
[5] I Pop T Grosan and R Cornelia ldquoEffect of heat generated byan exothermic reaction on the fully developed mixed convec-tion flow in a vertical channelrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 471ndash4742010
[6] E M Sparrow G M Chrysler and L F Azevedo ldquoObservedflow reversals and measuredpredicted nusselt numbers fornatural convection in a one-sided heated vertical channelrdquoJournal of Heat Transfer vol 106 no 2 pp 325ndash332 1984
[7] M M S El-Din ldquoFully developed forced convection in avertical channel with combined buoyancy forcesrdquo InternationalCommunications in Heat and Mass Transfer vol 19 no 2 pp239ndash248 1992
[8] A J Chamkha ldquoOn laminar hydromagnetic mixed convectionflow in a vertical channel with symmetric and asymmetric wallheating conditionsrdquo International Journal of Heat and MassTransfer vol 45 no 12 pp 2509ndash2525 2002
[9] A J Chamkha T Grosan and I Pop ldquoFully developed mixedconvection of a micropolar fluid in a vertical channelrdquo Inter-national Journal of Fluid Mechanics Research vol 30 no 3 pp251ndash263 2003
[10] T Grosan R Pop and I Pop ldquoThermophoretic deposition ofparticles in fully developedmixed convection flow in a parallel-plate vertical channelrdquo Heat and Mass Transfer vol 45 no 4pp 503ndash509 2009
[11] E Magyari ldquoThermophoretic deposition of particles in fullydeveloped mixed convection flow in a parallel-plate verticalchannel the full analytical solutionrdquo Heat and Mass Transfervol 45 no 11 pp 1473ndash1482 2009
[12] A Barletta ldquoLaminar mixed convection with viscous dissipa-tion in a vertical channelrdquo International Journal of Heat andMass Transfer vol 41 no 22 pp 3501ndash3513 1998
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mechanical Engineering
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Advances inAcoustics ampVibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspJournalthinspofthinsp
Sensors
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Antennas andPropagation
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mechanical Engineering
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Advances inAcoustics ampVibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
thinspJournalthinspofthinsp
Sensors
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Antennas andPropagation
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
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