Heat PorousSphere Final 2

8/10/2019 Heat PorousSphere Final 2

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Studies on flow through and around a porous permeable sphere: II.

Heat Transfer

A. K. Jain and S. Basu1

Department of Chemical EngineeringIndian Institute of Technology Delhi

New Delhi 110016, India

ABSTRACT

Heat transfer from a porous-permeable sphere due to flowing of fluid through and around

it is studied for a wide range of Reynolds number (0.02 to 2000) and Pradtl number (0.7

to 7) using standard CFD software. CFD simulated results for impermeable sphere

predicts the Whitaker correlation quite well. The Simulation is then extended for heattransfer from an isothermal porous-permeable sphere. The results are presented in terms

of four dimensionless parameters - particle Reynolds number (Re) based on the free

stream velocity and diameter of the porous sphere, permeability ratio, Pradtl number (Pr)

and Nusselt number (Nu). The results show that the heat transfer rate from porous

permeable sphere increases with the increase in permeability. At low Re, the Nu for a

permeable sphere is higher than that for solid sphere when the Pr is low, whereas, for

high Pr, permeability has only a weak effect on the Nu. At high Re, the Nu for permeable

sphere is much higher than that for solid sphere irrespective Pr values. The correlation

obtained from the CFD simulation data for heat transfer from porous permeable sphere is

useful in predicting Nu for porous permeable sphere for a wide range of Re from 0.02 to

2000 and Pr from 0.7 to 7 at different permeability ratios.

Keywords:Porous permeable sphere, Heat transfer, Permeability Ratio

INTRODUCTION

Heat transfer from an object due to flowing of surrounding fluid is a topic of industrial

importance. The object concerned may be relatively simple, such as a cylinder or sphere,

or it may be more complex, such as a tube bundle made up of a set of cylindrical tubes

with a stream of gas or liquid flowing between them. In the present investigation,

spherical aggregates forming porous permeable sphere is considered because of

1Corresponding author; e-mail: [email protected]; Fax +011 91 26581120

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simplicity in analyzing symmetric flow condition, availability of data and for the purpose

of comparing results for solid sphere. The results may be applicable to many processes

that involve spray of bubbles or droplets (Bird et al., 2002). Also, it may be useful in

understanding the geothermal energy transportation under varied geological structures

(Vasudeviah and Balamurugan, 1998).

Acrivos and Taylor (1962) studied the problem of forced convection from an isothermal

sphere for small and large Peclet number (Pe). Their analysis is valid when Reynolds

number (Re) and Pe ( = Re. Pr) is less than one, with no restriction on the Prandtl number

(Pr). Different empirical heat transfer correlations were given by different investigators

for single impermeable sphere along with experimental data e.g., Ranz and Marshall

(1952), Whitaker (1972), Achenbach (1978) and Romkes et al. (2003). The different

empirical correlations are given below.

Ranz and Marshall (1952)

31

21

PrRe66.00.2 +=Nu for 3.5 < Re < 7.6 x 104 (1)

Whitaker (1972)

32

32

21

PrRe06.0Re4.00.2

++=Nu (2)

for 3.5 < Re < 7.6 x 104

and 0.7 < Pr < 380Achenbach (1978)

++= 6.14 Re10*34

10.2Nu for 10

2< Re < 2 x 10

5 (3)

It is well known fact that the Nusselt number (Nu) is equal to 2 for a sphere immersed in

an infinite medium as the steady state conduction solution prevails (Whitaker, 1972).

Johnson and Smet (1984) examined the heat transfer from a permeable sphere in uniform

flow and at low Re. They considered the case when conduction is dominant heat transfer

mechanism in the exterior fluid, i.e. small Pe, and the case when convection was

dominant, i.e. large Pe. They found from their study that at small Peclet numbers the heat

transfer rate from a permeable sphere differ at leading-order from that found for an

impermeable sphere, whereas, for large Peclet numbers permeability has only a weak or

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second-order effect on the heat transfer rate. But their theoretical work was only limited

to Stokes flow. They did not study the system for high Reynolds number.

In the present work heat transfer study on flow through and around a porous permeable

sphere for a wide range of Re (0.02 to 2000) and Pr (0.7 to 7) is studied using standard

CFD software. The main objective is to study the effect of permeability and to examine

how the heat transfer process is influenced by fluid flow not only around but also through

the porous permeable sphere. A generalized correlation is suggested based on all

simulation results obtained for a wide range of Reynolds number, Prandtl number and

permeability ratio.

COMPUTATIONAL FLUID DYNAMICS SIMULATION

Porous permeable sphere is assumed to consist of small spherical particles (grains) with

some specific void fraction. Mass and energy conservation expressions and the Navier-

Stokes equations are used for the outer region of porous permeable sphere. Whereas, in

the inner region, mass and energy conservation and Darcys law of Brinkmans extension

are applied in order to solve the temperature, pressure and velocity fields. Thus, the flow

regions through and around a porous permeable sphere are divided into two parts,

namely, internal flow and external flow. The flow regions and the coordinate systems for

flow through and around a porous permeable sphere are shown in figure 1 of Jain et al.

(submitted).

External flow:

Equation of continuity, Reynolds-average Navier-Stokes (RANS) given in Jain et al.

(submitted) were used along with the energy equation. The energy equation is given

below.

Equation of energy

( ) ( )( ) ( ) heffjjj

eff SvJHTPEvEt

+

+=++

(1)

Where E is total energy, is the density, P is the pressure, T is the temperature, H is the

enthalpy, effis the effective shear stress, Jj is the diffusion flux of speciesjand effis the

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effective thermal conductivity. eff = + Cp t / Prt, where Prt is the turbulent Prandtl

number, Cpis the specific heat of fluid and is the thermal conductivity of the flowing

fluid. The first three terms on the right-hand side of eq. (9), represent energy transfer due

to conduction, species diffusion, and viscous dissipation, respectively. Sh includes the

heat of chemical reaction, and any other volumetric heat sources have been defined. In

the present study Shis zero.

Internal flow:

(i) Momentum equations for porous media:

Porous media are modeled by the addition of a momentum source term to the Reynolds-

average Navier-Stokes equation. The source term is composed of two parts, a viscous

loss term and an inertial loss term. For simple homogeneous porous media, the source

term, the Kozeny equation for relating relating permeability and porosity, the inertial

resistance factor, Darcys equation and intertial loss term given in Jain et al. (submitted)

were used for the modeling of internal flow region. It should be noted that isothermal

condition is assumed inside of the porous permeable sphere.

Meshing of the system and setting of tolerance limit was done following same procedure

as described in Jain et al (submitted). Laminar model was used for low Reynolds number

and k- model is used at high Reynolds number. Inside the porous permeable sphere

laminar model is used.

Calculation of heat transfer coefficient (h) and Nusselt number (Nu):

The temperature of the surface of the particle, constitutes the porous permeable sphere,

was set to a surface temperature, Ts, whereas the fluid at the boundary of the domain

(inlet condition) was set to a lower bulk temperature, T. As a result of the temperature

difference between the particle surface and the fluid, heat is transferred from the particle

to the fluid. The particle-to-fluid heat transfer rate, q, for flow around sphere of diameter

d, and heat transfer coefficient, h, is expressed as, q = h d2 (Ts-T) and Nusselt number

is expressed as, Nu = h d / .

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RESULT AND DISCUSSION

Solid sphere:

At first, simulations are carried out for solid sphere for different Reynolds number

ranging from 0.02 to 2000 and different Prandtl number varying from 0.7 to 7. The

simulation results are plotted in the form of Whitaker correlation (Whitaker, 1972) and

compared with the experimental data. In figure 1, the symbols show experimental data

whereas dashed line indicates the Whitaker correlation (eq. 1). It is well known fact that

the correlations available in the literature (Holman, 1986) slightly under predict the

experimental value for Nu at low Re. It is seen in figure 1 that the experimental data are

well predicted by the CFD simulations, as shown by solid line at low Re. The simulated

Nu values for high Reynolds number are also in good agreement with the experimental

data available in literature.

Porous permeable sphere:

In figure 2, Nu, is plotted against Re Pr2/3

at different permeability ratio, K/d2for flowing

of air through and around the porous permeable sphere. The solid line indicates the Nu

for impermeable sphere whereas different dashed lines indicate Nu for permeable sphere

of different K/d2values. Nu for solid sphere approaches to the asymptotic value of 2 with

the decrease in Re. Nu increases with the increase in Re for all K/d2values. The increase

in heat transfer coefficient with the increase in Re is due to the increase in velocity field

inside the porous permeable sphere, which in turn increased the heat transfer rate. The

increase in Nu with K/d2

values for porous permeable sphere is significant for high

Reynolds number. The heat transfer coefficient increases with the increase in K/d2value

because of the lower resistance to flow offered by highly permeable sphere. At low

Reynolds number, value of Nu for permeable sphere approaches to that for solid sphere.

Present simulation result on Nu for porous permeable sphere is verified with the

analytical solution given by Johnson and Smet (1984) at low Re (Table 1). They

concluded that the Nu for porous permeable sphere is higher than that for solid sphere at

low Pe ( = Re. Pr) of 0.015. On the other hand, Nu has weak dependence on permeability

at a slightly higher Pe (say, 0.14) and hence heat transfer rate from permeable and solid

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sphere is almost the same. It is seen in table 1 that the Nu for porous permeable sphere

differs by 15 % from that for impermeable sphere at low Re and Pr. However, Nu for

porous permeable sphere differs only by 0.09 % from that for impermeable sphere at low

Re and high Pr. Further, at high Re, permeability has strong effect on Nu irrespective of

the Prandtl number. Thus, Nu for porous permeable sphere is higher than that for solid

sphere at high Reynolds number. The Nu increases with the increase in Pr for both

permeable and impermeable sphere at high Re, whereas no significant increase in Nu is

with the increase in Pr at low Re.

Based on simulated results a correlation for heat transfer from porous permeable sphere is

generated using non-linear regression analyses. The regression coefficient of 0.91 was

obtained. The correlation for heat transfer from porous permeable sphere is given by,

( ) ( )1.13

21

23

22

1

1PrRe7.32

+=

dKNu (15)

valid for, 0.02 < Re < 2000, 0.7 < Pr < 7 and 10-5

< K/d2< 10

-2

When predicting the Whitaker correlation (for solid sphere) using above equation (eq.

15), an average error of 15 % was obtained at low Re with zero permeability value (K/d2

= 0). This correlation can be extended for mass transfer from porous permeable sphere

since the phenomena of mass transfer is analogous to heat transfer. In eq. (15), Nu and Prhave to be replaced by Sherwood number and Schimdt number. This would work fine for

no change in size of porous permeable sphere due to mass transfer.

CONCLUSION

Heat transfer study on flow through and around a porous permeable sphere is investigated

with the help of standard CFD software for wide ranges of Reynolds number and Prandtl

number. Experimental data and Whitaker correlation (1972) for heat transfer from solid

sphere are excellently predicted by present CFD simulation. The CFD simulation then

extended for heat transfer from isothermal porous permeable sphere by considering mass,

momentum and energy equations outside of the porous permeable sphere and Darcys

law with Brinkmans extension inside the porous permeable sphere. Present simulated

results show that the heat transfer rate increases with the increase in permeability. At low

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Re, the Nu for a permeable sphere is higher than that for solid sphere when the Pr is low,

whereas, for high Pr, permeability has only a weak effect on the Nu. At high Re, the Nu

for permeable sphere is much higher than that for solid sphere irrespective Pr values. The

correlation obtained from the CFD simulation data for heat transfer from porous

permeable sphere is useful in predicting Nu for porous permeable sphere for a wide range

of Re from 0.02 to 2000 and Pr from 0.7 to 7 at different permeability ratios.

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Notation

Cp specific heat of fluid, J kg-1

K-1

d diameter of sphere, m

E total energy, N m

h heat transfer coefficient, Wm-2

K-1

H enthalpy, N m

Jj diffusion flux of species j, mole m-2

s-1

K permeability, m2

K/d

2

permeability ratio, dimensionless

Nu Nusselt number (= h D / ), dimensionless

P pressure, N/m2

Pe Peclet number (= Re. Pr), dimensionless

Pr Prandtl number (= Cp/ ), dimensionless

Prt turbulent Prandtl number, dimensionless

q heat transfer rate, Watt

Re Reynolds number (= d v / ), dimensionless

Sh heat source term in eq. 1

t time, s

T temperature, K

Ts sphere temperature, K

T bulk temperature, K

vi velocity component in xidirection

vj velocity component in xjdirection

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vl velocity component in xldirection

v approaching velocity, m/s

Greek letters

thermal conductivity, W m-1K

eff effective thermal conductivity, W m-1

K

viscosity of fluid, Pa s

t turbulent viscosity of fluid, Pa s

density of fluid, kg/m3

eff Effective shear stress, N m-2

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References

Achenbach, E., in: Proceeding of the 6th

international heat transfer conference on heat

transfer from sphere up to Re = 6 x 106, Vol. 5, Hemisphere 1978 Washington DC.

Acrivos, A., and T. D. Taylor, Heat and mass transfer from single sphere in stokes flow,

J. Phys. Fluids, (1962) 5(4), 387-394.

Bird R. B., Stewart E. W., Lightfoot N. E., Transport Phenomena, John Wiley and

Sons 2002, India.

FLUENT Users Guide, Fluent Inc., 1998.

Hinze, J. O., Turbulence, McGraw-Hill, 1975 New york.

Holman, J. P., Heat transfer, McGraw-Hill, 1986 New York.

Jain, A. K., C. Sirker, and S. Basu, Studies on flow through and around a porous

permeable sphere: I. Hydrodynamics (Submitted)

Johnson, R. E., and R. P. Smet, On the heat transfer from a permeable sphere in stokes

flow, Chem. Eng. Sci., (1984) 22(7) 947-958.

McCabe L. W., and C. J. Smith, Unit operation of chemical engineering, 3rd

edition,

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McGraw-Hill Kogakusha, Ltd, 1980, Japan.

Ranz, W. E., and W. R. Marshall, Evaporation from drops, Chem. Eng. Prog., (1952) 48,

141-146.

Romkes, S. J. P., F. M. Dautzenberg, and C. M. van den Bleek, and H. P. A. Calis, CFD

modeling and experimental validation of particle-to-fluid mass and heat transfer in a

packed bed at very low channel to particle diameter ratio, Chem. Eng. J., (2003) 96, 3-13.

Vasudeviah, M., and K. Balamurugan, Heat transfer from a porous sphere in a slow

viscous flow, Int. J. Non-linear Mechanics, (1998) 33(1), 111-124.

Whitaker S., Forced convection heat transfer correlations for flow in pipes, past flat

plates, single cylinders, single spheres, and for flow in packed beds and tube bundles,

AIChE J. (1972) 18 (2), 361-370.

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Figure captions

Fig. 1. Comparison of simulated results with Whitaker (1972) correlation and

experimental results for heat transfer from impermeable sphere

Fig. 2. Forced-convection heat transfer from a porous permeable sphere

Table caption

Table 1: Nusselt number values for flow through and around sphere at different Reynolds

number

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Fig. 1. Comparison of simulated results with Whitaker (1972) correlation

and experimental results for heat transfer from a impermeale sphere

Re

0.01 0.1 1 10 100 1000 10000

(Nu-2)/Pr

0.4

0.01

0.1

1

10

100

Whitaker correlation

(Nu-2)/Pr0.4

=0.4Re0.5

+0.06Re2/3

Experimental

Present simulation

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Fig. 2. Forced-convection heat transfer from a porous permeable sphere

Re Pr2/3

0.001 0.01 0.1 1 10 100 1000 10000

Nu

1

10

100

1000

k/d2= 1e-2

k/d2= 1e-3

k/d2= 1e-4

k/d2= 1.82e-5

solid sphere

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Table 1: Nusselt number values for flow through and around sphere at different Reynolds

number

PecletNumber(Pe=Re. Pr)

Reynoldsnumber(Re)

Prandtlnumber(Pr)

Nu for permeablesphere(K/d

2= 1e-2)

Nu forimpermeablesphere

(K/d2= 0)

Variationbetween Nu forpermeable and

impermeable

sphere

0.0149 0.02 0.7442 2.5371 2.2062 14.99 %

0.1398 0.02 6.9909 2.2428 2.2408 0.09 %

1488 2000 0.7442 265.7559 26.4973 902.95 %

13980 2000 6.9909 2410.7440 69.6281 3361.32 %

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Heat PorousSphere Final 2