Entropy generation in the Blasius flow under thermal radiation

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Phys. Scr. 85 (2012) 035008 (6pp) doi:10.1088/0031-8949/85/03/035008

Entropy generation in the Blasius flowunder thermal radiationAdnan Saeed Butt1, Sufian Munawar1, Asif Ali1 and Ahmer Mehmood2

1 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan2 Department of Mathematics, FBAS, International Islamic University, Islamabad 44000, Pakistan

E-mail: adnansaeedbutt85@gmail.com

Received 29 September 2011Accepted for publication 26 January 2012Published 27 February 2012Online at stacks.iop.org/PhysScr/85/035008

AbstractIn this work, we examine the effects of thermal radiation and viscous dissipation on entropygeneration in the Blasius flow. The governing boundary layer equations for the velocity andtemperature fields are transformed into ordinary differential equations with the help of suitablesimilarity transformation and then solved numerically with the help of the shooting method.The expressions for the volumetric entropy generation rate and the Bejan number arecalculated for thermal radiation. The effects of various physical parameters on the entropyproduction number, average entropy and the Bejan number are studied through graphs usingvelocity and temperature profiles. It is noticed that by increasing the thermal radiationparameter the entropy production in a thermal system can be reduced and viscous dissipationincreases the entropy production.

PACS number: 44.20.+b

1. Introduction

The Blasius flow is widely recognized as the boundary layerflow induced over a resting flat plate by a uniform freestream. The study of moving flows over a flat plate has beenof great importance for engineers and scientists because ofits fundamental role in the fields of aerodynamics, chemicalengineering, mechanical engineering, etc. In 1908, Blasius [1]presented a solution to the problem in the form of a powerseries. Howarth [2] made computations by hand of the Blasiusflow using the Runge–Kutta numerical method to reanalyzethe problem. The problem was reinvestigated by Abussita [3]and the validity of the solution was established. Several otherstudies have been conducted on the solutions of the Blasiusflow [4–8].

The effects of thermal radiation on the boundarylayer flow are very important in processes involving hightemperatures, for example, gas turbines, nuclear powerplants, thermal energy storage, etc. For this reason, Hossainet al [9, 10] studied the effects of thermal radiation, using theRosseland diffusion approximation, on free convection alonga porous vertical plate. Raptis et al [11] analysed radiativeflows in the presence of magnetic field. Cortell [12] studiedthe radiation effects in a Blasius flow.

In thermodynamic analysis of flow and heat transferprocesses, one thing of core interest is to improve the thermalsystems in order to avoid energy losses and fully utilizethe energy resources. The thermodynamic irreversibility

associated with a system can be quantified through entropygeneration in the system, which is a useful tool for predictingthe performance of engineering processes by investigatingthe irreversibility arising during the processes. Entropygeneration is a measure of the destruction of the availablework of the system. Various sources such as heat transfer dueto solid boundaries and viscous dissipation are responsiblefor the generation of entropy [13, 14]. Entropy generationin flow systems was initially investigated by Bejan [15].The author showed that the engineering design of a thermalsystem could be improved through minimization of entropyproduction. Sahin [16] introduced the second law analysis ofa viscous fluid in a circular duct with isothermal boundaryconditions. Yilbas et al [17] studied entropy generation in asemi-blocked pipe including the swirling effect. Arpaci [18]studied the entropy production due to radiation. The effect ofviscous dissipation on the entropy generation due to naturalconvection from a heated horizontal isothermal cylinder inoil has been investigated by Abu–Hijleh et al [19]. Mahmudand Fraser [20, 21] applied the second law analysis to basicconvective heat transfer problems in non-Newtonian fluidflow through a channel made of two parallel plates. Al-Odatet al [22] studied the effect of magnetic field on the entropygeneration due to laminar forced convection past a horizontalflat plate.

In this paper, the effect of thermal radiation on entropygeneration in the Blasius flow is investigated. This studycan be useful in analysing the processes involving high

0031-8949/12/035008+06$33.00 Printed in the UK & the USA 1 © 2012 The Royal Swedish Academy of Sciences

Phys. Scr. 85 (2012) 035008 A S Butt et al

temperatures where the contribution of thermal radiationis of vital importance, for example, cooling systems forelectronic devices, the cooling of nuclear reactors, thermalheat exchangers, etc. The information obtained can beemployed to design thermal systems with reduced sources ofirreversibilities. A numerical technique known as the shootingmethod is used to obtain the solution, and graphs are plottedfor the entropy generation number, averaged entropy andthe Bejan number for different values of various parameters.A brief discussion of the solution and graphs and theconclusions are presented.

2. Mathematical formulation of the problem

For the classical Blasius flat-plate flow problem, the governingequations of motion for the flow and heat transfer in thepresence of thermal radiation and viscous dissipation can besummarized by the following boundary value problem:

∂u

∂x+

∂v

∂y= 0, (1)

u∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2, (2)

u∂T

∂x+ v

∂T

∂y=

k

ρcp

∂2T

∂y2+

ν

ρcp

(∂u

∂y

)2

−1

ρcp

∂qr

∂y, (3)

subject to the boundary data

u = v = 0 at y = 0,

u → U∞ as y → ∞,(4)

T = Tw at y = 0,

T = T∞ as y → ∞,(5)

where u and v are the velocity components along the flowdirection (x-axis) and normal to the flow direction (y-axis),respectively. ν is the kinematic viscosity, k the thermalconductivity, cp the specific heat of the fluid at a constantpressure, ρ the density, qr the radiative heat flux, Tw thetemperature of the wall, T∞ the temperature of the ambientfluid and U∞ is the constant free stream velocity. Using theRosseland approximation for radiation [11], the radiative heatflux can be simplified as

qr = −4σ1

3k1

∂T 4

∂y, (6)

where σ1 and k1 are the Stefan–Boltzmann constant and themean absorption coefficient, respectively. It is assumed thatthe temperature differences within the flow are sufficientlysmall such that the term T 4 can be expressed as a linearfunction of temperature. This is accomplished by expandingT 4 in a Taylor series about the free stream temperature T∞

and neglecting higher-order terms. These computations leadto the following approximation:

T 4= 4T 3

∞T − 3T 4

∞. (7)

Using equations (6) and (7) in the last term of equation (3),we obtain

∂qr

∂y= −

16σ1T 3∞

3k1

∂2T

∂y2. (8)

Introducing this in equation (3), we obtain the followingenergy equation:

u∂T

∂x+ v

∂T

∂y=

(α +

16σ1T 3∞

3ρcpk1

)∂2T

∂y2+

ν

ρcp

(∂u

∂y

)2

, (9)

where α =k

ρcpis the thermal diffusivity. It can be seen from

equation (9) that the presence of radiation enhances thethermal diffusivity.

We introduce the following similarity transformations tonormalize equations (1)–(5):

u = U∞ f ′(η), v =1

2

√U∞

νx(η f ′

− f ), η = y

√U∞

νx.

(10)Substituting (10) into (2), we have

f ′′′ +1

2f f ′′

= 0. (11)

The corresponding boundary conditions become

f = 0, f ′= 0 at η = 0,

f ′→ 1 at η → ∞.

(12)

For the prescribed surface temperature, we introduce thedimensionless temperature θ(η)

θ(η) =T − Tw

T∞ − Tw. (13)

Using the above transformation, equation (9) and theboundary conditions (5) can be written as

θ ′′ +1

2Pr

(3NR

3NR + 4

)f θ ′ + Pr Ec

(3NR

3NR + 4

)f ′′2

= 0 (14)

and

θ = 1, at η = 0,

θ → 0, as η → ∞,(15)

where Pr =µcp

k is the Prandtl number, Ec =U∞

cp(Tw−T∞)is the

Eckerd number and NR =kk1

4σ1T 3∞

is the radiation parameter.

2.1. Entropy generation equation

The local volumetric rate of entropy generation SG for aviscous fluid in the presence of thermal radiation is definedas [18]

SG =k

T 2∞

[(∂T

∂y

)2

+16σ1T 3

∞

3k1

(∂T

∂y

)2]

+µ

T∞

(∂u

∂y

)2

.

(16)Clearly, equation (16) shows the contribution of the three

main sources of entropy production. The first term on theright-hand side is the entropy generation due to heat transfer,the second term is the local entropy generation due to thermalradiation, whereas the third term is the entropy generation dueto the fluid friction. It is appropriate to define a dimensionlessnumber Ns for the entropy generation as

Ns =SG

SG0=

1

Rex

[(1 +

4

3NR

)θ ′2 +

Pr Ec

�f ′′2

], (17)

2

Phys. Scr. 85 (2012) 035008 A S Butt et al

Figure 1. Effect of the radiation parameter on the local entropygeneration number.

Figure 2. Effect of the Prandtl number on the local entropygeneration number.

where SG0 =k(Tw−T∞)U 2

∞

T 2∞

ν2 , � =T∞

Tw−T∞

and Rex =U∞x

νare

the characteristic entropy generation rate, the dimensionlesstemperature difference and the local Reynolds number,respectively. From the above expression for the entropygeneration number, the averaged entropy generation numbercan be evaluated using the following integral formula:

[Ns]avg =1

∀

∫∀

Ns d∀, (18)

where ∀ is the length of the boundary layer region. Anotherimperative irreversibility parameter is the Bejan number Be,which is the ratio of entropy generation due to heat transfer tothe entropy generation:

Be =Entropy generation due to heat transfer

Total entropy generation. (19)

When Be � 0.5, the irreversibility due to heat transferdominates, whereas when Be � 0.5 the irreversibility due tothe viscous effect dominates. When Be = 0.5, heat transferand fluid friction entropy generation are equal.

3. Results and discussion

The nonlinear ordinary differential equations (11) and (14)along with boundary conditions (12) and (15) are solved

Figure 3. Effect of the Eckert number on the local entropygeneration number.

Figure 4. Effect of � on the local entropy generation number.

numerically using the shooting method with the help of thesymbolic software Mathematica by keeping the accuracy goalequal to 10−20. The semi-infinite domain is truncated ata suitable distance where the effects of both the types ofboundary layers are negligible. To see the physical effectsof different parameters involved in the governing equationson the entropy production, graphs of the local entropygeneration number, the averaged entropy generation numberand the Bejan number are plotted in figures 1–10. A detaileddiscussion of the graphical results is given in the followingsubsections.

3.1. Entropy generation analysis

Figure 1 depicts the effects of the radiation parameter NR

on the local entropy generation number. It is observed thatby increasing the radiation parameter, entropy generationdecreases. This behaviour of entropy generation is only dueto the increase of the absorption rate of radiation. Hence,for large values of NR, the entropy generation rate is small.Therefore, it is the best way of increasing the thermal radiationparameter in order to reduce entropy production. Also it isnoticed that for a particular value of NR, the local entropygeneration number is maximum at the plate surface. Thisshows that the surface of the plate is a major source ofentropy generation as the heat transfer and fluid friction aremore prominent at the surface and in the region close to

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Phys. Scr. 85 (2012) 035008 A S Butt et al

Figure 5. Effect of the radiation parameter on the average entropygeneration number.

Figure 6. Effect of the Eckert number on the average entropygeneration number.

Figure 7. Effect of the radiation parameter on the Bejan number.

it. The entropy generation number decreases as one movesin the downstream direction and approaches zero where theeffects of the boundary layer are negligible. The effect of thePrandtl number Pr on Ns is shown in figure 2, which showsthat an increase in Pr results in an increase of the entropygeneration rate near the surface. However, as one moves awayfrom the surface, an inverse behaviour is observed in entropygeneration. Hence the entropy generation rate is small for thefluid having a low Prandtl number. Figure 3 presents the effectof the Eckerd number Ec on the local entropy generation

Figure 8. Effect of the Prandtl number on the Bejan number.

Figure 9. Effect of the Eckert number on the Bejan number.

Figure 10. Effect of the dimensionless temperature difference onthe Bejan number.

number and it is seen from this that entropy generationincreases with increasing Ec. Therefore, it is concluded thatthe loss of worthless energy can be reduced by controllingthe penetration of viscous dissipation. The variation of theentropy generation number with respect to the dimensionlesstemperature difference � is shown in figure 4. The value ofNs decreases with an increase in the value of �, which is dueto the diminishing of the difference between the temperatureof the surface and of the ambient fluid.

Figure 5 shows the variation of the averaged entropygeneration number [Ns]avg against the Prandtl number Pr

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Phys. Scr. 85 (2012) 035008 A S Butt et al

for different values of the radiation parameter. The reasonfor choosing the Prandtl number as the varying parameter isits practical importance and strong contribution to entropygeneration. For a particular value of NR, it is noticed thatthe average entropy is an increasing function of Pr . So theirreversibility effect increases as the Prandtl number increases.Moreover, for a fixed Pr the average entropy decreasesas the radiation parameter increases. Figure 6 presents theeffects of the Eckert number Ec on the averaged entropygeneration number [Ns]avg plotted against the Prandtl number.The averaged entropy generation increases with an increase inthe Eckert number.

3.2. Fluid friction versus heat transfer irreversibility

The local entropy generation number Ns is a good tool forgenerating the spatial entropy profile, but fails to give anyidea of whether the fluid friction irreversibility dominatesor the heat transfer. For this purpose, usually the Bejannumber Be is examined by plotting its expression givenin equation (19). In figure 7, the effects of the radiationparameter NR on the Bejan number are presented. With anincrease in the radiation parameter NR, the fluid frictionirreversibility becomes dominant near the surface of theplate because of high absorption rate. However, heat transferirreversibility increases as one moves away from the surface.It is observed that the effects of heat transfer irreversibilityare immense and continue to increase far away in the flowregime due to the presence of thermal radiation. Figure 8shows the effects of the variation of Pr on the Bejan number.It is seen that irreversibility due to fluid friction is dominantwithin the boundary layer region and irreversibility effectsdue to heat transfer rate dominate in the free stream region.Figure 9 demonstrates the effects of the Eckert number Econ Be. As the value of Ec increases, the irreversibility dueto fluid friction becomes strong in the boundary layer region.The Bejan number for various values of the dimensionlesstemperature difference � is plotted in figure 10. Fromthe figure, it is observed that heat transfer irreversibilitydominates as � increases. The overall behaviour of the Bejannumber in the spatial domain illustrates that near the plate heattransfer irreversibility has a strong effect, then away from theplate fluid friction irreversibility becomes strong and in theambient fluid heat transfer irreversibility dominates.

3.3. Minimization of entropy generation

From an engineering point of view, the main purpose ofentropy generation analysis of complex thermal systems isto identify the methods of minimizing the irreversibilityassociated with different physical parameters in order toobtain the optimized range of parameters. This enablesengineers to design thermal systems with small energy losses,and thus minimizing the entropy generation helps maintain themaximum available energy for use. From the above entropyanalysis, it is observed that by controlling certain physicalparameters, the optimal design and efficient performance ofengineering systems can be enhanced. It is clear from figures 1and 5 that the production of entropy can be minimized bychoosing a high radiation parameter. Hence the high valueof the thermal radiation parameter prevents the loss of useful

energy. Moreover, from figures 2, 5 and 6 it is quite obviousthat the small Prandtl number is associated with small entropy.Therefore, to minimize entropy production in a thermalsystem, fluids with small Pr are recommended. It is alsoobserved that the large Eckert number corresponds to largeentropy production. Hence to avoid this deficiency, one shoulduse fluids that have small viscous forces or develop methodsin which fluid friction effects can be reduced.

4. Conclusions

In the present study, the effects of thermal radiation andviscous dissipation on entropy generation in the Blasius floware investigated. The influence of these parameters on theentropy generation number and Bejan number is discussed. Itis observed that there is an increase in the entropy generationnumber Ns with increasing Prandtl number Pr and Eckertnumber Ec. On the other hand, the effects of the radiationparameter NR on Ns are decreasing. It is worth noting thatnear the surface and in the region close to it, the irreversibilityeffects due to fluid friction are significant; however, theirreversibility due to heat transfer becomes dominant in themain stream region. From the analysis made above, it isalso observed that by choosing appropriate values of physicalparameters, entropy generation can be minimized.

Acknowledgment

We are grateful to an anonymous reviewer for expertcomments and suggestions for improving the discussion partof this paper.

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