This paper presents an analytical solution for the Graetz job extended to steal flow that includes rarefaction consequence. The hydrodynamically developed flow is assumed to come in a round microchannel with unvarying wall temperature. The consequence of speed faux pas, temperature leap and syrupy dissipation term are all considered. The consequence of nondimensional parametric quantities ( Knudsen figure, Prandtl figure, Brinkman figure ) on local and to the full developed Nusselt figure is investigated. The consequences show that under certain conditions the syrupy dissipation consequence on heat transportation in microchannels is important and should non be neglected.
Keywords: Microchannel, Graetz Problem, Slip flow, Syrupy dissipation, Heat transportation
In recent decennaries, advancement in micro fiction engineering has lead to development in a assortment of micro scale fluidic systems. Micro devices are by and large referred to devices holding a characteristic length graduated table between 1 millimeters and 1 m such as microchannel heat money changers, micropumps, micro actuators, etc. The extended technology applications of microdevices have provoked research workers to analyze on its fluid flow and heat transportation features.
Modeling heat and fluid flow through such little devices is different from the macro scales opposite numbers. As the ratio of the average free way to characteristic length ( Knudson figure, ) increases, the continuum premise becomes no longer valid [ 1,2 ] . For the rarefaction changing between 0.001 and 0.1 the government is called slip-flow government [ 3 ] . Under such fortunes the continuum patterning along with the speed faux pas and temperature leap boundary conditions on the wall demand to be considered.
The hydrodynamically developed and thermally developing laminal flow come ining a round tubing with changeless wall temperature known as Graetz job was analytically solved by Graetz [ 4,5 ] . Many research workers developed the job by sing different boundary conditions and different cross subdivisions or by including the axial conductivity and syrupy dissipation. By increasing the applications of microchannels, their ascendant boundary conditions were applied to work out Graetz job. Barron et Al. [ 6 ] extended the job by including the consequence of faux pas speed. Ameel et Al. [ 7 ] presented an analytical solution with changeless wall heat flux in round microchannel. Tunc and Bayazitoglu [ 8 ] solved the job by sing slip speed, temperature leap and syrupy dissipation consequence with unvarying wall temperature and unvarying heat flux boundary conditions. Jeong and Jeong [ 9 ] solved the energy equation with syrupy dissipation and axial conductivity footings in analogue home bases sing both unvarying temperature and unvarying heat flux boundary conditions. Cetin et Al. [ 10 ] presented a numerical solution by sing syrupy dissipation and axial conductivity with changeless wall temperature in microchannel with round cross subdivision. Discrepancies in the to the full developed Nusselt Numberss can be seen in the consequences of Tunc and Bayazitoglu [ 8 ] ( analytical solution ) and Cetin et Al. [ 10 ] ( numerical solution ) . Cetin et Al. [ 11 ] besides solved the Graetz job analytically with changeless wall heat flux in microtubes including the consequence of syrupy dissipation and axial conductivity.
ratio of specific heats,
thermic adjustment coefficient
average free way ; eigenvalue
distance along tubing
dimensionless distance along tubing,
speed in ten way
faux pas flow
local heat transportation coefficient
local Nusselt figure
This paper extends the Graetz job to include the rarefaction consequence and syrupy dissipation term in the fluid with changeless wall temperature boundary status. The energy equation is analytically solved by the method of separation of variables. By specifying appropriate nondimensional variables the energy equation becomes a well-known differential equation which is known as Kummer. The influence of syrupy dissipation on Nusselt figure in both instances where the fluid is being cooled or heated is exhaustively discussed.
The steady province hydrodynamically developed flow with inlet temperature T0 enters into a microtube with radius r0 and wall temperature. The ruling energy equation may be written as:
Boundary conditions are:
where is the temperature of the fluid at the wall, the wall temperature, is the thermic adjustment coefficient and is the specific heat ratio. The to the full developed speed profile can be derived from impulse equation by using faux pas speed boundary status:
Appropriate non-dimensional variables may be defined as:
By using non-dimensional variables and replacing Eq. ( 3 ) into Eq. ( 1 ) , the energy equation can be written as:
As so, hence Eq. ( 6 ) becomes
The to the full developed temperature profile can be derived from Eq. ( 8 ) by using Eq. ( 7a ) and Eq. ( 7c ) as boundary conditions:
By replacing Eq. ( 10 ) into Eq. ( 6 ) we obtain:
By using the method of separation of variables we have:
The solution for Eq. ( 12 ) is:
By specifying the undermentioned transmutation:
Eq. ( 13 ) may be written as:
Eq. ( 16 ) is known as Kummer equation and the solution is:
Where is written in the signifier of following series:
By replacing into Eq. ( 7c ) we obtain:
By work outing Eq. ( 19 ) we can happen.
Therefore the solution for the temperature distribution may be written as:
As the Eq. ( 13 ) is Sturm-Liouville equation and the eigenfunctions are extraneous with regard to burdening map, by using Eq. ( 7b ) can be written as:
The mean temperature in the round tubing can be written as:
Puting the equation ( 22 ) into non-dimensional signifier, the dimensionless bulk temperature may be written as:
The heat flux at the wall may be written as:
by seting Eq. ( 24 ) into non-dimensional signifier we obtain:
The local Nusselt figure can be written as:
3. RESULTS AND DISCUSSION
All the computations have been carried out by presuming and. In Figure 1 the consequence of Prandtl figure on to the full developed Nusselt figure in different Knudsen figure is presented. It is shown that by increasing Prandtl figure the to the full developed Nusselt figure additions. When, the fluctuation of Prandtl figure does non hold consequence on Nusselt figure. It is besides apparent that in lower Prandtl Numberss to the full developed Nusselt figure alterations more by fluctuation of rarefaction.
Figure. Variation of to the full developed Nu as a map of
Pr for different Kn and Br=0
In Figure 2 the fluctuation of to the full developed Nusselt figure by different rarefactions with and without syrupy dissipation is given. It is seen that by increasing Knudsen figure the to the full developed Nusselt figure lessenings and it has more decreasing consequence when syrupy dissipation is taken into history.
In Figure 3 the solid lines shows the fluctuation of Nusselt figure in the length of tubing at. It is clear that by increasing Knudsen figure the to the full developed Nusselt figure will diminish. By sing the influence of syrupy dissipation, it can be seen that a leap occur in Nusselt profile and the to the full developed Nusselt figure additions. Neglecting syrupy dissipation, the to the full developed Nusselt figure lessenings from 3.656 to 3.069 by increasing Knudsen from 0 to 0.1 that is approximately 16 % lessening in Nusselt figure while in the presence of syrupy dissipation the to the full developed Nusselt figure decreases about 50 % .
Figure. Variation of to the full developed Nu as a map of Kn with and without
syrupy dissipation ( Pr=0.8 )
Figure. Variation of the local Nu as a map of dimensionless axial co-ordinate
for different Kn and Br ( Pr=1 )
The consequence of Brinkman figure on Nusselt figure is depicted in Figure 4. It is found that the Nusselt figure in developing part has greater values for greater Brinkman Numberss. It is shown that the to the full developed Nusselt figure additions by sing the consequence of syrupy dissipation. When the Brinkman figure is non equal to zero which means the syrupy dissipation is present, fluctuation of Brinkman figure does non hold consequence on to the full developed Nusselt figure.
Figure. Variation of the local Nu as a map of dimensionless axial co-ordinate for
different Br ( Kn=0.1, Pr=1 )
Figure 5 illustrates the fluctuation of Nusselt with different Brinkman Numberss. Fully developed Nusselt figure reaches a same value for both positive and negative Brinkman Numberss. When Brinkman is negative the fluid is being heated and hence there is a location in the length of tubing that the majority temperature is equal to the wall temperature. There exists a remarkable point where Nusselt figure goes to the eternity.
Figure. Variation of the local Nu as a map of axial co-ordinate for positive and
negative Br ( Kn=0.4, Pr=1 )
Table 1 and Table 2 show the to the full developed Nusselt figure for different values of Knudsen, Prandtl and Brinkman.
The to the full developed Nusselt figure ( Br = 0 )
The to the full developed Nusselt figure ( Br = 0.01 )
In this survey an analytical solution for the Graetz job in microchannels with round cross subdivision is presented. Syrupy dissipation consequence is taken into history. It is found that syrupy dissipation causes a leap in Nusselt profile and the to the full developed Nusselt figure additions. In the presence of syrupy dissipation fluctuation of Brinkman figure does non hold consequence on to the full developed Nusselt figure. It can be concluded that under certain conditions the influence of syrupy dissipation on Nusselt figure is important and should non be neglected.