Analyses and Modeling of Laminar Flow in Pipes Using Numerical Approach

doi:10.4236/jsea.2012.59076

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Journal of Software Engineering and Applications, 2012, 5, 653-658

http://dx.doi.org/10.4236/jsea.2012.59076 Published Online September 2012 (http://www.SciRP.org/journal/jsea)

653

Analyses and Modeling of Laminar Flow in Pipes Using

Numerical Approach

O. Saheed Ismail, George T. Adewoye

Department of Mechanical Engineering, University of Ibadan, Ibadan, Nigeria.

Email: [email protected], [email protected]

Received June 7th, 2012; revised July 13th, 2012; accepted July 22nd, 2012

ABSTRACT

This paper investigate some important works done on numerical analysis and modeling of laminar flow in pipes. This

review is focused on some methods of approach and the analytical tools used in analyzing of the important parameters

to be considered in laminar flow; such as frictional losses, heat transfer etc. in laminar flow in pipes of different shapes,

and the importance of laminar flow in its areas of applications. Prominent researchers have approached this from dif-

ferent perspectives. Some carried out analysis on the pressure drop as a function of permeability, some worked on fric-

tion factor analysis, some discussed heat transfer effects of laminar flow in the entrance region, while some discussed

its applications in various industries. Some of these works were done considering a given form of pipe configuration or

shape which is circular pipes. Only a few, of the literature reviewed have related their considerations to different forms

of pipes. Most consider pipes to be majorly circular in shape, but in industries today some circular pipes have become

elliptical in shape due to long time usage of the pipes, which would have contributed to increase in some different forms

of losses in the industries. In engineering, efficiency and effectiveness improvement is the major goal, if a research

work has been done, considering the important parameters in laminar flow showing their effects on different forms of

pipe configuration as a result of pipe deformation due to usage, huge amount of money will be saved. This will show

clearly how the efficiency of a given circular pipe has seriously been affected due to deformation, and the level of loss

this has resulted to.

Keywords: Laminar Flow; Flow Parameters; Circular and Elliptical Pipes Flow; Numerical Modeling and Analyses

1. Introduction

Laminar flow is a gentle flow in which the streamlines

are not crossing each other, that is, they are parallel to

one another. What determines a flow if it is laminar in

nature or not is the value of its Reynolds number, if the

Reynolds number is less than 2000, the flow is still con-

sidered to be laminar flow. The laminar flow still re-

mains an important form of flow in engineering. A flow

in engineering can be compressible flow or incompressi-

ble flow. The incompressible flow finds its applications

in the area of pipe flow in which the pipe length may be

too short for achieving fully developed conditions, such

as in a short length heat exchangers, the incompressible

flow has its density remains constant. While in the in-

compressible region the flow parameters changes with

temperature change and this may result to a significant

drop in the pressure. When the pressure drop due to the

flow of the gas is large enough, causing a considerable

decrease in density, then the flow may be considered to

be compressible, and appropriate formulas that take into

consideration changes in both density and velocity must

be used to describe the flow. The compressible flow

finds its application in the area of increase in capacity of

natural gas transport and distribution networks, which

depends on gas pressure loss [1]. Proper investigation of

this kind of flow has brought a great development to both

science and technology, due to its wide range of applica-

tions in these fields.

As the fluid flows through a pipe many things happen,

heat is generated and the flow pressure also reduces, this

is due to the friction existing between the flow fluid and

the wall of the pipe. Estimation of the value of the head

loss hl is very important for proper engineering design.

One important formula for calculating the value of head

loss hl is given by Darcy-Weisbach. This is shown in

Equation (1) below:

hDg

 (1)

hl is given as the head loss, f represents the Darcy fric-

tion factor, L stands for the length of pipe, V represents

the flow velocity, D is the internal diameter of the pipe

Analyses and Modeling of Laminar Flow in Pipes Using Numerical Approach

654

and g represents the gravitational.

In the study of external flow over a body, the relation-

ship between the wall heating and the change of skin

friction drag, which is caused by the difference in viscos-

ity and density of a fluid when it is heated, can easily be

seen to be proportional to the temperature ratio taken to

the power of −2/3. On the other hand, simple calculations

on the momentum equation of incompressible gas flow

through a pipe show that for a constant pressure drop, the

mass flow rate is a function of inflow temperature taken

to the power of −2.5. This means that by increasing the

inflow temperature, the mass flow rate will be decreased

considerably (Kramer et al., 1999 cited in [1]).

Analysis of fluid flow subjected to heat addition is also

very important. Rayleigh in the late 19th century fol-

lowed by Fanno in the early 20th century Carried out an

analytical solution on a compressible pipe flow subjected

to heat addition, they also made an attempt to combine

this with friction loss [2]. But for effective transfer of

heated fluid through a pipe with no significant loss of

heat, a heat pipe is good for this kind of fluid transporta-

tion. A concentric annular heat pipe (CAHP), as shown

in Figure 1, consists of two concentric pipes of different

diameters attached to each other by means of end caps,

which creates an annular vapor space between the two

pipes. Wick structures are placed on both the inner sur-

face of the outer pipe and the outer surface of the inner

pipe. Concentric annular heat pipes (CAHP) are more

effective than conventional heat pipes and can be used in

many applications including energy conversion systems,

cooling of diesel engine pistons etc. (Bankston et al.,

1971 cited in [3])

All these analyses of the flow parameters really play a

vital role in the industries to avoid unnecessary expenses,

and boost the production processes. In the area of design,

it helps to ensure accuracy and reliability of design, be-

cause it assists in making proper assumptions and proper

selection of the design parameters.

2. Some Approaches to Laminar Flow

Analyses

Several researchers have approached the analyses of

laminar flow with different views. Soundalgekar et al.

[4], analyzed laminar flow in porous circular pipe by

perturbation theory, in which flow parameters such as

axial and radial velocity profile, skin friction, axial pres-

sure and mass flow were considered. They discovered

that skin friction decreases with increase in pressure gra-

dient, while the mass flow increases with increase in

pressure drop.

Zhao et al. [5] also generated a numerical solution to a

laminar forced convection in a heated pipe subjected to a

reciprocating flow. Findings include, the four major heat

transfer parameters associated with this problem, which

are; the kinetic Reynolds number, Rω, the oscillation am-

plitude, Ao, the length to diameter ratio (L/D), and the

Prandtl number of the fluid. The research shows that the

average heat transfer rate increases with both the kinetic

Reynolds number, Rω, and oscillation amplitude, Ao, but

decreases with length to diameter ratio (L/D). The pres-

sure drop in a flowing fluid through a channel with a

porous wall is a function of wall permeability, channel

dimension, axial position, and fluid properties [6].

Flow through the annular region is an important form

of flow in the study of fluid mechanics; its behavior

cannot be over emphasized, because the behavior of fluid

flow through the annular region has its great application

in the industries, majorly oil industries, therefore, know-

ing the losses encounter with this kind of flow will help

in the effective design of this kind of unit in the indus-

tries. With the help of Computational Fluid Dynamics

Figure 1. Schematic of a CAHP and coordinate system (adapted from [3]).

Analyses and Modeling of Laminar Flow in Pipes Using Numerical Approach 655

(CFD), viscoplastic fluid flows in annular spaces, focus-

ing on the parameters such as; the profiles of pressure

drop, entrance length, axial and tangential velocities, and

on the flow path prediction has been thoroughly investi-

gated. These variables are usually considered relevant for

an understanding of well drilling mudflow and the parti-

cles transported by it [7].

Mass and heat transfer are also investigated on a

laminar flow of water over an ice layer which is sub-

jected to slip condition. To analyze this problem a para-

metric mathematical model is used to simulate the cou-

pled heat and mass transfer events occurring in moving

boundary, problem associated with a quasi-steady state

steady flow process [8]. Vapour flow is also analyzed in

a concentric annular heat pipe numerically using a simple

algorithm. Navier-stoke equation was used to simulate

the fluid flow and the heat transfer in the annular vapour

space, and the governing equations are solved numeri-

cally using finite volume approach. The results from this

were compared with the previous research works and it

shows a strong agreement in accuracy [3].

Nouri-Borujerdi et al. [9] carried out an analysis that

predicts the critical mass flow rate, pressure, vapor qual-

ity, and void fraction in a capillary tube under critical

condition using a drift flux model. By using the dimen-

sional analysis by Buckingham’s π theory, they came out

with some generalized correlation for predicting the flow

properties as a function of the flow parameters and the

capillary tube sizes under various critical conditions.

This research work was done using the following pa-

rameters ranges; the inlet pressure in the ranges of 0.8 ≤

Pinlet ≤ 1.5 Mpa, the subcooling temperature in the range

of 0 ≤ ΔTsub ≤ 10˚C, the tube diameter is in the range of

0.5 ≤ D ≤ 1.5 mm and tube length is in the range of 1 ≤ L

≤ 2m for water, ammonia, refrigerants R-12, R-22 and R-

134 as working fluids. This study is of a great importance

in the design of refrigeration system.

Behaviour of gas flowing in pipes is another important

area of fluid flow investigation. Compressible flow in gas

pipeline that was subjected to wall friction and heat

transfer was numerically modeled. The result obtained

from this study was investigated on a natural gas pipeline

under different thermal condition. How the heat affects

the pressure drop, temperature, and the match number

were also investigated. These are caused by the friction

and the heat exchange changes [2]. Numerical analysis

was carried out on fluids flow through a smooth circular

micro-channels with different diameter, two different

fluids were considered; air with slip boundary conditions

and water with no slip boundary conditions. In this case

the dissipation term in the energy equation was consid-

ered due to the small dimensions of the micro-channel.

The momentum equation was combined with the energy

equation and solved using a finite volume approach. The

results show that with the air as flowing fluid the pres-

sure gradient is nonlinear along the micro-channel, while

considering water as the flowing fluid and considering

adiabatic boundary condition, the smaller the micro-

channel the higher the velocity of flow, and an increase

in the flow temperature would be observed, and the tem-

perature increase is a linear function of the axial position

[10].

Laminar flow has also been analyzed in micro-chan-

nels with a partial semi-circular profile. This work inves-

tigated fully developed laminar flow as a function of the

circularity index, ĸ, which is the ratio of the radii along

the curved surface to the radii along the flat surfaces of

the partial semi-circular profile. A correction factor, K, to

the Hagen-Poisuille could be determined, and was related

to the circularity index by Equation (2);

2.56

5.299KK (2)

It was observed that, the level of wall shear stress,

when normalized by the pressure drop per unit length,

increased approximately linearly with increase in the

circularity index, K [11]. Yang et al. [12] study fully de-

veloped flow in a curved pipe with arbitrary curvature

ratio (the ratio of the pipe radius to the pipe curvature),

but in their study they also put heat transfer into consid-

eration by heating the pipes. In this study the finite dif-

ference numerical method is used to solve the full Na-

vier-stokes for the modeled problem. Unlike, the previ-

ous studies which considered the curvature ratio in the

ranges less than 0.3, the curvature ratio was varied in

ranges from 0.1 to 0.9, while both the Reynolds Pradtl

number were varied from 1 to 2000, and 0.7 to 300 re-

spectively. The result of this study shows that friction

ratio and the Nusselt number ratio show a good correla-

tion with the curvature ratio parameters, the Dean num-

ber (De) and the pradtl number. Another important ap-

proach, which is very effective for the analysis of heat

flux in the flowing fluid is boundary element method

(BEM), which has a better advantage compared to finite

difference or finite element method due to the fact that

instead of full domain discretization, only the boundary

is discretized into elements and internal point position

can be freely defined. Therefore the quantity of data

necessary to solve the problems can be greatly reduced

(Brebbia et al. 1984 cited in [13]). Laminar heat convec-

tion problem between two coaxial cylinders with con-

stant heat flux boundary condition were analyzed using

dual reciprocity boundary element method (DRBEM).

This kind of numerical approach to fluid flow and heat

transfer between two coaxial cylinders under different

number of boundary elements has given a strong accu-

racy compared to other numerical tools [13].

Heat transfer in fluid flow has also been considered by

focusing on the effect of fins which contributes to the

Analyses and Modeling of Laminar Flow in Pipes Using Numerical Approach

656

rate of heat transfer. In this study effects of internal fins

were investigated as a function of number of fins present

and as a function of the height of the fins. This study was

done for both laminar and turbulent flow. For laminar

flow it was discovered that the mean Nusselt number

decreases and the friction factor increases as the number

of fins increases, while for turbulent flow, both the mean

Nusselt and friction factor increase. But, for increase in

the height of the fins led to increase in friction factor for

all Reynolds number selected [14].

Most of the research works have concentrated much

on fluid flow in circular pipes, but the research study that

considered fluid flow in another important form of pipe

configuration, that is, elliptic cylindrical pipe, was done

by [15]. In this study, fluid flow in elliptic cylindrical

was investigated in the hydrodynamics entrance region.

They studied various flow parameters and came out with

some reliable results, which can be used to describe the

behavior of laminar flow in an elliptic cylindrical pipe.

The nature of fluid flow in circular pipe with that elliptic

cylindrical pipe, and they came out with some facts

which are helpful in the industries today.

3. Tools for the Analytical Study and

Simulation of Laminar Flow in Pipes

In the study of laminar flow in pipes several analytical

tools have been used to model and analyze the behavior

of laminar flow in pipes Some of the tools are very good

in the analyses of some specific parameters in the flow

fluid, while some other tools are very effective in the

analyses of some other flow parameters.

Mehran et al. [16] used Artificial Neural Network

(ANN) to estimate the friction factor in pipe. Artificial

Neural Network (ANN) is used to solve the Colebrook-

White equation. The Colebrook-White equation provides

a relationship between the Reynolds number, relative

roughness and the Darcy friction factor. Unlike the pre-

vious method for solving the Colebrook-white, the Arti-

ficial Neural Network (ANN) is less time consuming and

accurate. [3] made use of finite volume approach to solve

the Navier-Stokes equations in order to analyze the be-

havior of a vapor flow in a concentric circular annular

pipes. In this study they discretized the governing equa-

tion using finite volume approach and they only consid-

ered the axial symmetric by the annular vapour space in

their numerical analysis.

CFX 3D (version 4) is another package used in solving

the governing equation numerically of a particular flow

problem, and it will also analyze the results. CFX 3D is a

commercial simulator for numerical resolution of prob-

lems involving fluid mechanics and heat transfer. It uses

the methodology of finite volumes with structured me-

shes, and is sufficiently flexible to enable the study of

complex geometries through the use of generalized coor-

dinates [15]. Besides, there is the possibility of resolution

employing a multigrid methodology. This renders the

software appropriate for scientific research, making pos-

sible to test different mathematical models and to analyze

the influence of certain parameters, without requiring the

arduous work of numerical implementation of the Na-

vier-Stokes equations or mathematical models as quoted

by [15].

Viscous dissipation in micro-channels has also been

studied numerically using SIMPLEC algorithm as an

analytical tool, with the aid of Buckingham π theory as

cited in [10]. [17] simulated laminar flow in pipes in two

dimensions using lattice Boltzmann equation (LBE) with

multiple-relaxation-time (MRT) collision model. In con-

trary to the usual computational fluid dynamics (CFD)

which makes use of the direct discretization of the Na-

vier-Stokes equations, the lattice Boltzmann equation

(LBE) is derived from the Boltzman equation and kinetic

theory. It is this kinetic origin of this lattice boltzman

equation (LBE) makes it so much different from the

usual computational fluid dynamics (CFD), cited by [17].

[18] produced an asymptotic solution to unidimensional

axisymmetric laminar flow in pipe which is subjected to

sudden retardation or acceleration. Non-integer order of

derivatives was also used as a tool for modeling a lami-

nar liquid flow in pipes with frequency-dependent fric-

tion. This tool was used to model correctly hydraulic

transmission lines in order to foresee the limitation in

functions of such transmission lines [19]. A compact

approximation model was proposed to predict the pres-

sure drop in a variety of shapes for a fully developed,

laminar, incompressible flow in smooth mini and mi-

cro-channels. This model is a function of the geometrical

parameters, that is, cross-section, which are; the area,

perimeter, and the polar moment of inertial. This model

shows a very good agreement when compared with the

numerical and analytical solution of several shapes [20].

Spectral-homotopy analysis method and a novel suc-

cessive linearization method have been used to analyze

two-dimensional incompressible laminar flow in a rec-

tangle domain bounded by two permeable surfaces. The

spectral-homotopy analysis method and the novel suc-

cessive linearization method were used to solve the gov-

erning equation, which is, nonlinear fourth-order differ-

ential equation. These two methods have been computa-

tionally proved that they are efficient and reliable for

solving problems related to nonlinear boundary value

[21]. Pseudo spectral collocation method is used to cal-

culate the first order perturbation quantities in analyses

of flow in a channel whose walls describe a traveling

wave motion. From this analysis, the position of flow

separation and reattachment can be easily located, and

also the variation in the velocity and pressure with fre-

Analyses and Modeling of Laminar Flow in Pipes Using Numerical Approach 657

quency of excitation can be determined [22].

4. Application of Laminar Flow

Laminar flow has been of a very good important form of

flow, this has made its application in most of the science

and engineering field to be very useful.

Laminar air-flow systems have been used by the aero-

space industry to control particulate contamination such

as dust or lint which could affect the reliability of preci-

sion parts [23]. Laminar flow cabinets were also used to

control the level of micro-organisms in a given environ-

ment that is to achieve an environment that is free of mi-

cro-organisms; the air flowing in such environment must

be laminar in nature [23]. In the field of Microbiology

laminar flow is applied generally in three major areas,

which are:

 Product protection: In this case laminar flow is ap-

plied in the environment such that the product protec-

tion is the main goal, that is, where critical sterility

and assays are needed, whereby the personnel in such

environment is less considered in the design of the

laminar air-flow in such environment, that is, the

personnel protection is not a problem, Standard hori-

zontal laminar flow clean benches can be used for the

procedures.

 Personnel protection: This is an environment where

some infectious materials are been processed, in this

case for the safety of the technical personnel a verti-

cal laminar flow cabinet can be used.

 Personnel and Product protection: When the mi-

crobial contaminants have to be hindered and the per-

sonnel have to be protected also, a vertical laminar

flow cabinet has to be used [23].

It was also discovered that laminar airflow used in an

operating room can yield mean levels of viable airborne

contamination at critical sites as low as 0.05 organisms

per cubic foot of air [24]. Chen Hua-de et al. found that

laminar air flow can be useful in the treatment of burns,

by sampling the air in the laminar flow chamber and the

wound tissues of the patients for bacteria detection. From

their results they concluded that the application of lami-

nar air flow to a serious burn can assist in the treatment

of the burn, even within a shorter treatment time [25].

5. Circular Pipe and Elliptical Pipe;

Laminar Flow Behaviour

Most of the research works focused their attention on the

behavior of laminar flow on circular pipes, as less con-

sideration has been given elliptical pipes, which is also

very important. Circular pipes may deform to an ellipti-

cal shape after a series of usage due to temperature and

pressure of fluid it conveys over a given period of time,

the behavior of flow in the pipe at this time should be

clearly known to the engineers. That is how elliptic cy-

lindrical pipe contributed to more loss in industries

unlike the circular pipes, and the effect of this should be

stated so that, time value of a given pipe can be techni-

cally determined, and this will help us to understand

when a given pipe is due for replacement.

However, as some of the important properties of lami-

nar flow were not considered in comparison of laminar

flow behavior in circular pipe and elliptic cylindrical

pipe, [15] still illustrated some comparison between cir-

cular and elliptical cylindrical pipes in their research

work, these comparisons were enumerated below:

 To achieve the same mass flow rate in both the circu-

lar and elliptic cylindrical pipe, of the same cross-

section area, the value of the pressure in the circular

pipe has to be increased by almost 50% in the elliptic

cylindrical pipe so as to achieve the same mass flow

rate as in the circular pipe.

 In circular pipe the effective wall to wall distance is

the same as the hydraulic diameter, Dh. While in el-

liptic cylindrical pipe the effective wall to wall dis-

tance is over estimated by the hydraulic diameter, Dh.

Therefore, this fact of mismatch between the hydrau-

lic diameter, Dh, and the effective wall-to-wall dis-

tance, explains the reason why the circular pipe as an

higher aspect ratio compared to elliptic cylindrical

pipe.

6. Conclusion

This work has revealed some important applications of

laminar flow in pipes, and has also shown that most

analyses carried out on laminar flow in pipes were ma-

jorly on circular pipe, while less consideration has been

given to elliptic cylindrical pipe, which is another im-

portant pipe configuration in industries. More so, it has

clearly shown how fluid flow is adversely affected by a

deformed circular pipe into an elliptic cylindrical shape,

by considering the behavior of some of the important

flow parameters in these two different pipe configura-

tions. This work also shows that some more important

flow parameters such as heat transfer etc. are to be con-

sidered when analyzing the flow in these two different

pipes configuration. This will help the transportation of

fluid through circular pipes in industries, by understand-

ing the enormous loss that results, when circular pipes

are deformed due to usage over time and not been re-

placed on time.

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