Modelling and Theoretical Analysis of Laminar Flow and Heat Transfer in Various Protruding-Edged Plate Systems

Laminar flow and heat transfer in different protruding-edged plate systems are modelled and analyzed in the present work. These include the Parallel Flow (PF) and the Counter Flow (CF) protruding-edgedplate exchangers as well as those systems being subjected to Constant Wall Temperature (CWT) and Uniform Heat Flux (UHF) conditions. These systems are subjected to normal free stream having both power-law velocity profile and same average velocity. The continuity, momentum and energy equations are transformed to either similarity or nonsimilar equations and then solved by using well validated finite difference methods. Accurate correlations for various flow and heat transfer parameters are obtained. It is found that there are specific power-law indices that maximize the heat transfer in both PF and CF systems. The maximum reported enhancement ratios are 1.075 and 1.109 for the PF and CF systems, respectively, at Pr = 100. These ratios are 1.076 and 1.023 for CWT and UHF conditions, respectively, at Pr = 128. Per same friction force, the CF system is preferable over the PF system only when the power-law indices are smaller than zero. Finally, this work demonstrates that by appropriately distributing the free stream velocity, the heat transfer from a plate can be increased up to 10% fold.


Introduction
Conversion and utilization of energy often involve heat transfer process.This process is encountered in many engineering applications.These applications include steam generation and condensation in power plants; sensible heating and cooling of viscous fluids as in thermal processing of pharmaceutical, agricultural and hygiene products; evaporation and condensation of refrigerants in refrigeration and air-conditioning systems; cooling of engine and turbomachinery systems; and cooling of electrical appliances and electronic devices.It is well known that improving heat transfer over that in the typical practice results in significant increases in both the thermal efficiency and the economics of the plant operation.Improving heat transfer is a terminology that is frequently referred to it in the literature as heat transfer enhancement or augmentation [1].
Heat transfer enhancement mechanisms basically reduce the thermal resistance in a conventional thermal system by promoting higher convective heat transfer coefficient that can be accompanied with surface area increase.Consequently, the size of a thermal system can be reduced, or the heat duty of an existing thermal system can be increased, or the pumping power requirements can be reduced [1]- [4].These enhancement mechanisms are classified into passive and active methods [3].Of special interest to this work is the passive enhancement method.These methods are primarily comprised of at least one of the following mechanisms: (a) increasing the surface area [5]; (b) interrupting the boundary layer to promote the convective heat transfer coefficient [6]; (c) using of liquid-vapor phase change [7]; (d) using the surface coatings to increase velocity near boundaries [8] [9]; (e) using the liquid and gas additives to enhance thermophysical properties [6] [10]; (f) using the flow rate and velocity amplification devices [11] [12]; and (g) layering the immiscible flows [13]- [15].In the present work, it is interested to investigate heat transfer enhancement due to properly distributing a given flow rate before striking a plate having a protruding edge.This protruding edge is physically important to ensure one-directional stagnation flow along the plate so that heat transfer is maximized.
When a normal free stream strikes a plate having a protruding edge at its inlet, stagnation flow occurs along the plate length which has its stagnation line coinciding with the plate inlet edge.This flow is characterized by having an increasing axial velocity in the vicinity of the plate from zero at the inlet to maximum at the exit [16]- [18].In addition, it is characterized by having decreasing normal velocity from maximum at the free stream to zero at the plate.Allowing for most of the normal free stream flow rate to be near the inlet causes increases in both axial and normal velocities closing to the plate inlet which promote the average convective heat transfer coefficient.On the other hand, the heat transfer rate is expected to decrease when most of the normal free stream flow rate is considered to be near the plate exit.It is because the latter effect results significantly suppressing the local convective heat transfer coefficients in the upstream region while slightly promoting these coefficients downstream.It is therefore expected that there may be a specific normal free stream velocity profile that can maximize the heat transfer rate from a plate having a protruding edge at its inlet.To the author best knowledge, this proposal has not been investigated in the literature and accordingly it is considered as the motivation behind the present work.
In the next section, the geometries of various analyzed systems composed of plates with protruding edges are explained.These systems include the Parallel Flow (PF) and the Counter Flow (CF) protruding-edged plate exchangers.These systems are exposed to normal free stream having both power-law velocity profile and same average velocity.The continuity, axial momentum and energy equations of the fluids adjacent to the plate are transformed to either similarity and nonsimilarity equations.Also, various similarity equations are obtained for protruding-edged plates subjected to either constant wall temperature or uniform heat flux conditions.The governing equations are solved numerically and are validated against well-established special cases.Different accurate correlations for flow and heat transfer parameters are obtained.An extensive parametric study has been conducted in order explore the influence of power-law index, Prandtl numbers and relative Reynolds numbers on Nusselt numbers and different heat transfer enhancement indicators.

Problem Formulation
The proposed two types of protruding-edged plate heat exchangers are shown in

Modeling of Laminar Flows in the Fluid Volumes in Vicinity of the Plate
Consider that the normal streams approaching the faces of the protruding-edged plate have the following velocity profile along the face length L : where h x and c x are the axial distances of the hot and cold fluids from the plate entrances, respectively, as shown in Figure 1  V are the average free stream normal velocities of the hot and cold fluids, respectively.m is the power-law index for both normal streams.The conservation of mass principle requires that the free stream axial velocities for the hot and cold fluids be equal to: where fluid and the plate and that between the normal free stream of the cold fluid and the plate, respectively.The dimensionless continuity and axial momentum equations of the hot and cold fluids in vicinity of the plate are given by [16] [17]: , , , 0

d h c h c h c h c h c h c h c h c h c
where x , y , and

Re
are given by: , , , where h ρ and h µ are the density and dynamic viscosity of the hot fluid, respectively.Those for the clod fluid are c ρ and c µ , respectively.The boundary conditions are given by: ( ,0 0

The Similarity Equations for the Laminar Flow in Vicinity of the Plate
Define the following independent and dependent variables: Equations 4(a), 4(b) are transformed to the given similarity equations when Equations ( 8) and ( 9) are used: The transformed boundary conditions are equal to: The average skin friction coefficient denoted by f C is equal to:

The Energy Equation for the PF and CF Systems
If h θ and c θ are defined as , , then, the energy equations of the hot and cold fluids are given by (Bejan, 2013):

c h c h c h c h c h c h c
where c h x x = for the PF system and 1 for the CF system.The heat transfer rate between the hot and cold fluids per unit width denoted by q′ can be computed from the following equation: Define the enhancement ratio λ as the ratio of the heat transfer rate to that quantity when 1 m = − .when 1 m = − , the flow in vicinity of the plate surface becomes no more stagnation flow and it will be an external flow parallel to flat plate.Mathematically, λ is equal to:

The Similarity Energy Equation for the PF System
Invoking the similarity variables given by Equations ( 8) and ( 9), Equations ( 15) and ( 16) for the PF system reduce to the following similarity equations and boundary conditions: ( ]

21(d)
The dimensionless heat transfer rate per unit width denoted by Θ is equal to the following for this case:

The Nonsimilarity Energy Equation for CF System
Invoking the following nonsimilarity variables: to Equations ( 14) and ( 15) for the CF system where 1 c h x x = − , the following nonsimilarity equations and boundary conditions are found: Θ for this case is equal to: The average skin friction coefficient , f HE

C
for the PF and CF systems is calculated from the following equation:

The Similarity Energy Equation for Constant Wall Temperature (CWT) Condition
Re → ∞ , the plate temperature approaches 1 c T .Thus, Equations ( 19)-( 21) reduce to the following: For this case, the local Nusselt number is defined as: ( ) where h h is the local convection heat transfer coefficient for the hot fluid flow.The average convective heat transfer coefficient h h given by can be computed from the average Nusselt number relation which is equal to:

The Similarity Energy Equation for Uniform Heat Flux (UHF) Condition
When the plate is generating uniform heat flux ( ) s q′′ at the surface facing the cold fluid, the dimensionless cold fluid temperature can be redefined as follows: , , This is in order to reduce the energy equation given by Equation 15(b) to a similarity equation.This similarity equation is given by: ( ] The boundary conditions for this case are given by: ( ) For this case, the local Nusselt number is defined as: The average Nusselt number relationship is given by: ( )

The Relation between Heat Transfer in PF and CF Systems and Nusselt Numbers
In terms of average convection heat transfer coefficients, the energy balance given by Equation ( 17) can be reduced to one equation given by: ( ) ( ) The definition of average Nusselt number can be used to show that Θ is equal to:

Numerical Methodology
Equation (10) was discretized using three points center differencing after substituting d d This re-sulted in having tri-diagonal system of algebraic equations, which was then solved using the Thomas algorithm [19].Iterations were implemented in the solution of Equation ( 10) because the second and third terms on the left of Equation ( 10) are non-linear.The following linearization models are used to linearize these terms [20]: ) ( ) = is solved using the trapezoidal rule [21].Note that the relationship between ( ) Also, Equations ( 19), ( 20), ( 29) and (34) were discretized using three points center differencing quotients and the resulted tri-diagonal system of algebraic equations have been solved using the Thomas algorithm without iterations.The left side of Equations 21(d) and 26(d) were discretized using two points difference quotients.
Under assumed plate temperatures, the solutions of the discretized forms of Equations ( 24) and (25) were obtained using the Thomas algorithm [19] and they were marched from , 0 , The marching procedure used for solving Equations ( 24) and (25) were repeated by letting the assumed plate temperatures ( )  equal to those modified by Equation (43).This process is continued until the maximum relative error between the assumed and modified plate temperatures is less than

Validations and Numerical Results
When 1 m = − and 0 m = , the flows become laminar flow parallel to flat plate and stagnation flow with uni- form normal free stream velocity, respectively.The solution for these two cases is well documented in literature [17].The comparisons between the present numerical method solutions and the reported values of the average Nusselt numbers for CWT condition when 1 m = − and 0 m = and the average Nusselt number for UHF con- dition when 1 m = − are shown in Table 1.Excellent agreements between both results are shown in this table.This lead to increased confidence in the results of the present work.

Accurate Correlations
Correlation for transformed axial velocity and the average skin friction coefficient ( ) f η ′ can be shown to be correlated to h η and m according to the following correlation: ( ) The coefficients ,

Correlations for maximum average Nusselt numbers and critical power-law indices
The maximum average Nusselt numbers for CWT and UHF conditions can be shown to be correlated to .These maximum values are obtained when the power-law index m is set to be equal to a critical value denoted by cr m .This critical value is correlated to the Prandtl number according to the following correlations: .

Correlations for exit Nusselt number and critical power law index for UHF condition
The maximum Nusselt number at the plate exit for the UHF condition can be shown to be correlated to

Discussion of Flow and Thermal Aspects for PF and CF Systems
Figure 9 shows that both hot and cold fluid temperatures increase as , h c η increase, respectively, for the PF protruding-edged plate exchanger.For the CF protruding-edged plate exchanger and as shown in Figure 10, the plate temperature is noticed to decrease as h ξ increases when 0 m < , while it increases as h ξ increases when 0 m > .When 0 m = , Equations ( 24) to (26) become similarity equations and physically PF and CF systems have same performance as the boundary layers at this case have fixed thicknesses for the CF system shows that the performance of the CF system can be accurately modeled by Equation (39) and the Correlation (48) for the UHF condition when 0. plots.Also, Figure 12 shows that the CF system has higher enhancement ratios than the PF system when m is quite below 0.2 m = − while the PF system has higher enhancement ratios than the CF systems when m is quite above 0.15 m = − . The heat transfer rates per same friction forces that is proportional to , f HE C Θ are seen in Figure 13 to be larger for the PF system than those for the CF system when 0 m > while it is vice versa when 0 m < .Also, this figure shows that

Conclusion
Laminar flow and heat transfer in various protruding-edged plate systems are modeled and investigated in the present work.These systems include the Parallel Flow and the Counter Flow protruding-edged plate exchangers as well as those systems being subjected to CWT and UHF conditions.These systems are exposed to normal free stream having both power-law velocity profile and same average velocity.The continuity, axial momentum and energy equations are transformed to similarity equations for CWT and UHF conditions as well as for the Parallel Flow system while they are transformed to non-similarity equations for the Counter Flow system.These equations are solved by using an accurate finite difference method.Excellent agreement is obtained between the numerical results and reported solutions of well-established special cases.Accurate correlations for different flow and heat transfer parameters are generated by using modern regression tools.It is found that there are always local maximum values for Nusselt numbers for both CWT and UHF conditions at specific power-law indices.Also, it is found that there are specific power-law indices that can maximize the heat transfer rate in the Parallel and Counter Flow systems.The maximum enhancement ratios for the Parallel and Counter Flow systems that are identified in this work are 1.075 and 1.109, respectively, which occur at Pr = 100.These ratios are 1.076 and 1.023 for CWT and UHF conditions, respectively, at Pr = 128.Per same friction force, the counter flow system is found to be preferable over the Parallel Flow system only when the power-law indices are smaller than zero.Finally, this work paves a way for new passive heat transfer enhancement method that can enhance heat transfer from a plate by a magnitude of 10% fold which is by appropriately distributing the free stream velocity.

Figure 1 .Figure 1 .
Figure 1.Schematic 2D diagram of the problem and the coordinates system for (a) Parallel Flow (PF) protruding-edged plate exchanger and (b) Counter Flow (CF) protruding-edged plate exchanger, and 3D isometric diagram for (c) PF and (d) CF systems.are at the plate entrances and they ban fluid flows in the opposite directions.In the CF system, the plate entrance of one face is opposing the entrance of the other face.Both entrances contain side protrusions so as to force the induced hot and cold fluid stagnation flows to have counter current directions as shown in Figure 1(b) and Figure 1(d).
h H and cH are the displacements between the normal free stream of the hot

1 cT
are the far stream hot and cold fluid temperatures, respectively.h Pr and c Pr are the Prandtl number for the hot and cold fluids, respectively.The boundary conditions are given by:

− 10 −
are the values of G and h f at the previous iteration, respectively.The values of 0.0005 and 6 were selected forh η ∆and the convergence criterion for the maximum relative difference in calculating G between two consecutive iterations.Next, the differential equation d d where M is the total number of discretized points along , h c ξ direction.N is the total number of discretized points along , .Then, the plate temperature was modified using the discretized form of Equation 26(d).This discretized equation can be rearranged in the following form: correlations have maximum relative error of 0.202% and 0.233% for the CWT and UHF conditions, maximum relative error of 0.0355% and 0.0309% for the CWT and UHF conditions, correlation has maximum relative error of 0.583% when 0.5 100 c Pr ≤ ≤ .The critical power-law index cr m that produces ,max L Nu is correlated to the c Pr according to the following correlation:

Figure 3 .
Figure 3. Effects of Pr h,c on

Figure 4 .Figure 5 .Figure 6 . 8 .Figure 7 .Figure 8 .
Figure 4. Effects of m on (8).It is shown in Figure11that the heat transfer rates between the hot and cold fluids for both PF and CF systems are maximized at critical power-law indices laying between 1 0 m − < < .The per- formance of the PF system is well modeled by Equation (39) and the Correlation (48) for the CWT condition as seen in Figure11on the plot given by This is because that the PF system has always constant separating plate temperature.The plot indented by in Figure12that the maximum heat transfer enhancement ratio is equal to max 1

Figure 9 .
Figure 9. Effects of m on temperature profile for PF system.

Figure 10 .Figure 11 .Figure 12 .
Figure 10.Effects of m on plate temperature for CF system.

Figure 13 .
Figure 13.Effects of m on

Figure 14 .
Figure 14.Effects of m on heat transfer enhancement ratio for CF and PF systems.
coefficient for heat exchanger, Equation (28)f dimensionless stream function, Equation(9)H displacement between the free stream and the plate[m] average) convection heat transfer coefficient [W•m −2 •K −1 ] L h local convection heat transfer coefficient at plate exit [W•m −2 •K −1 ] k fluid thermal conductivity [W•m −1 •K −1 ] L plate length [m]m power-law index for normal velocity profile, Equation (1) = q′ heat transfer rate per unit width [W•m −1 ] normal) fluid velocity in vicinity of the plate [m•s −1 ] o u axial free stream velocity at the plate exit [m•s −1 ] average) free steam normal velocity [m•s −1 ]

Table 2 .
The average Nusselt number for CWT and UHF conditions can be shown to be correlated to m and (a) Coefficients b i,j of the correlation given by Equation (44), i = 1, 2, 3, 4, 5; (b) Coefficients b i,j of the correlation given by Equation (44), i = 6, 7, 8, 9, 10.

Table 3
(a) and Table 3(b) for the CWT condition and

Table 4 (a) and Table 4(b) for
the UHF condition.These coefficients produce maximum relative error in computing