International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
Dimensional Analysis for Investigation of Effect of Air
Induction Pressure as an Operating Variable on a C.I. Engine
Performance
Dipak C. Talele #1, Dr. Dheeraj S. Deshmukh *2, Dr. Prashant P. Boranare #3
#1
Assistant Professor, Department of Mechanical Engineering, S.S.B.T.’s, College of engineering & Technology,
Bambhori, Jalgaon, India
*2
Professor & Head, Department of Mechanical Engineering, S.S.B.T.’s, College of engineering & Technology,
Bambhori, Jalgaon, India
#3
Assistant Professor, Department of Mechanical Engineering, S.S.B.T.’s, College of engineering &
Technology, Bambhori, Jalgaon, India
Abstract— Maximum efficiency and least possible
pollutant emissions from an internal combustion
engine are general objectives of performance analysis.
Air induction pressure as an operating parameter is
investigated to enhance Compression Ignition (C.I.)
engine performance. Engine performance is
dependent on fuel consumption, which bears direct
influence on efficiency and engine out emissions. Fuel
consumption rate is a basic dependent variable, It is
an important parameter of an engine, varies because
of any possible variable variations such as engine
design, its operating conditions and fuel, air induction
pressure and back pressure on engine that is caused
due to flow resistance offered by exhaust system
components. In this work, Dimensional analysis
technique is used for reduction of variables, for
analysis of effect engine operating variables and
model formulation thereby further optimization of
engine performance could be achieved.
Keywords-Air Induction Pressure, Efficiency,
Compression Ignition Engine, Performance
I. INTRODUCTION
In every system performance analysis, there is always
a need of comprehensive model development. For
representing the response in terms of proper
interaction of various inputs involved in any
phenomenon, the theory of experimentation is a good
approach. In fact it is felt that such an approach is not
yet seen towards research on I.C. engines. This
approach finally establishes an experimental data
based model for the phenomenon. Stationary class of
internal combustion engine includes mainly diesel
engines because of higher energy efficiency and
output power. In this complete process three crucial
steps viz. planning, operation and data analysis are
adopted [1]. In experimental planning step study of
instruments to be used for precision and accuracy
errors should be done. Due thought is given for
replication of readings against constant inputs for
better accuracy. Careful examination of the
independent and dependent variables, using
dimensional analysis for possible reduction in their
number is also carried out. Selected independent
ISSN: 2231-5381
variables are spaced and the proper sequence should
be followed in setting each point in turn.
The indicator diagram of a four-stroke diesel engine
cycle consists of two enclosed areas as shown in the
figure-1. The gross work done is represented by the
large area. The smaller shaded area represents the loss
of work due to exhausting of burnt gases and
admission of new unburnt gas or charge, formed by
the suction and exhaust operations is called pumping
loop. To obtain the net work done, work obtained
from the negative area is to be substracted from the
gross work. For justification purpose, the pumping
loop is magnified in figure. The volumetric efficiency
of the engine is affected by the gas exchange
processes. The performance of the engine depends on
the volumetric efficiency. when the piston moves from
bottom dead centre to top dead centre during the
exhaust stroke, pressure rises and gases are pushed
into exhaust pipe. Therefore the power required to
drive the exhaust gases is called the exhaust stroke
loss and increase in speed increases the exhaust stroke
loss. The indicator diagram of a diesel four-stroke
cycle engine shows the suction line ―ea‖ lies below the
atmospheric pressure line [7].
Owing to the restricted area, the entering air cannot
flow into the cylinder in sufficient quantity to keep the
pressure with the rapidly moving piston. The restricted
area of the inlet passages results in the fall of pressure
below the atmospheric pressure. Due to the restricted
area of the exhaust passages that do not allow the
gases to move out of the cylinder quickly as a result of
which the exhaust pressure remains somewhat higher
than the atmospheric pressure. Therefore the exhaust
line ―de‖ does not coincide with the atmospheric
pressure line but it rises slightly above it. With the use
of supercharger the air pressure of the inlet on I.C.
engine can be increased which results in decrease in
the negative loop of the indicator diagram of a fourstroke diesel engine cycle. This can increase the net
work done.
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
Figure 1: Theoretical Indicator diagram of a fourstroke diesel engine cycle
II. OPERATING VARIABLES FOR C.I. ENGINE
Operating parameters of an internal combustion
engine are reviewed and summary of important
operating variables is given below [2-6].
Table No1:-Operating parameters of an internal combustion engine
[2-6]
Sr.
No
Param
eters
Type of
variable
1
Fuel
consu
mption
rate
Basic
depende
nt
variable
2
Load
on
engine
Indepen
dent
variable
3
Speed
Indepen
dent
variable
4
back
pressur
e on
the
engine
Indepen
dent
variable
5
Air
inducti
on
pressur
e
Indepen
dent
variable
Causes of variations
It is a dependent & important Input
parameter of an engine varies
because of any possible variable
variations such as engine design
(type of engine) & fuel (type of fuel)
and After treatment system(type of
devices used and flow resistance
offered by exhaust system)
As per the output power requirements
such as vehicle weight, No. of
persons seating, road resistance or
slope condition, drag force because
of air while in motion.
As per the availability of time or time
requirements. Speed can be kept
constant as in case of electric
generator applications with the help
of a governor.
As per the complete exhaust system
designs and their maintenance
aspects because of aging effects.
Particulate Matter accumulations
causes variations of pressure drop
across each component of exhaust
system.
As per the design of air intake system
and its maintenance aspects because
of aging effects. Dust particles
accumulation in air filter causes
variations of pressure drop across
each component of induction system.
Turbocharger or Supercharger like air
induction system components also
affects on air induction pressure at
the inlet to C.I. engine.
ISSN: 2231-5381
III. REDUCTION OF VARIABLES
There are several quite simple ways to make a given
test compact in operating plan without losing
generality or control. The best known as well as most
powerful tool is dimensional analysis. In the past
dimensional analysis was primarily used as an
experimental tool whereby several experimental
variables could be combined to form one. Using this
principle modern experiments can substantially
improve their working techniques and be made shorter
requiring less time without loss of control. Deducing
the dimensional equation for a phenomenon reduces
the number of variables in the experiments. The exact
mathematical form of this dimensional equation is the
targeted model. Experimental data based modelling of
operating variables for determination of dimensionally
homogeneous equations is achieved by applying
Buckingham‗s Pi theorem. In this work for internal
combustion engine performance analysis almost all
the possible variables are considered. After critical
study of different variables only operating variables
for specific C.I. engine are selected for performance
analysis. Engine performance is basically dependent
on fuel consumption rate, which bears direct influence
on efficiency and engine out emissions. Fuel
consumption rate is a basic dependent variable. It is an
important parameter of an engine. It depends on all the
engine variables such as engine design, its operating
conditions, fuel composition and after treatment
system used. More efforts are required on the
development of the air induction system by further
study of the theory of operation in each component
used in air induction system of I.C. engines.
Future modifications must be done in such a way that
each alternation should not cause air induction fall.
Finally in this analysis the fuel consumption (Fc) of
four stroke, single cylinder C.I. engine during a test
run can be considered as dependent upon load(ld),
speed(N), back pressure(Pb) and suction pressure of
intake air (p1) on the engine. Using Buckingham`s Pi
theorem,
The fuel consumption (Fc) depends upon (i) ld, (ii) N,
(iii) Pb and (iv)P1 hence Fc is a function of ld, N and
Pb, P1. Mathematically,
Fc = f (ld, N, Pb, P1)
……. (I)
Or it can be written as
f1 (Fc, ld, N, Pb, P1) = 0
……… (II)
Therefore, total no of variables, n = 5
No of fundamentals dimensions, m = 3
(m is obtained by writing dimensions of each
variables as Fc = MT -1, ld = MLT -2, N=T -1, Pb =ML
-1
T -2, P1 = ML-1T-2. Thus, the fundamental dimensions
in the problems are M, L, T and hence m = 3).
Therefore the no. of dimensionless л – terms = n – m
=5–3=2
Thus, two л – terms say л1, л2 are formed.
Hence equation (II) is written as
f1(.л1, л2 ) = 0 ………………………..(III)
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
Each л – term = m + 1 variables
Where m is equal to 3 and also called repeating
variables. Out of five variables Fc, ld, N, Pb and P1,
three variables are to be selected as repeating variables,
Fc is a dependent variables and should not be selected
as repeating variables. Out of 4 remaining variables
(One variable should have geometric property, the
second variable should flow property and third one
should have fluid property. These requirements must
be fulfilled.) the variables ld, N and Pb are selected as
repeating variables. The variables themselves should
not form a dimensionless term and should have
themselves fundamental dimensions equal to m, i.e. 3
here. Dimensions of ld, N and Pb are MLT -2, T -1,
ML -1 T -2 and hence the 3 fundamental dimensions
exists in ld, N and Pb and they themselves do not
form dimensionless group.
Power of L,
Power of M,
Power of T,
b2= 0
Equating the powers of M, L, and T on both sides, we
get
Power of M,
0 = a1 +c1+1
a1 +c1 = -1 ………….….(i)
Power of L,
0 = a1 – c1…………… (ii)
Subtracting equation (ii) from (i),
c1 = -1/2 and by putting this value in equation (i), a1 =
-1/2
Power of T,
0 = -2 a1 –b1 -2 c1 -1
-1 = 2 a1 +b1 +2
c1 ………………… (iii)
Putting values of a1 & c1, we get
2 (-1/2) + b1+2 (-1/2) = -1
-1+b1 -1 = -1
b1 = -1 + 2 = 1
Substituting the values of a1, b1 and c1 in equation
(IV),
л1= ld -1/2. N 1. Pb -1/2. Fc
л1=
N .Fc
p .l
b
d
л2 = lda2N.b2 Pb.c2. P1
= Mo. Lo. To
-2 a2
= (MLT ) (T-1) b2 (ML-1T-2) c2 ML-1T-2
Equating the powers of M, L, and T on both sides, we
get
ISSN: 2231-5381
a2 = 0
c2= -1
0 = - b2 +2 -2
Substituting the values of a6, b6 and c6 in equation
(IV),
Л2 = ld0.N0. Pb.-1. P1
P1
Pb
Л2 =
л- terms in
Substituting the value of
equation (III),
f1(.л1, л2 ) = 0
f1(.
N .Fc
,
p .l
b
Each л – term is written as according to the equation
л1 = lda1N.b1 Pb.c1. Fc
a2 b2
л2 = ld N. Pb.c2. P1…………………. (IV)
л – terms are determined by the principle of
dimensional homogeneity.
For the л – term, we have
л1 = Mo. Lo. To
= (MLT -2) a1 (T-1) b1 (ML -1 T -2) c1
-1
MT
0 = a2 - c2 -1
0 = a2 +c2 +1
0 = -2a2 – b2 -2c2-2
or
d
N .Fc
p .l
p .l
Fc =
= f1(
d
b
or
P1
)=0
Pb
b
N
d
. f1(
P1
)
Pb
P1
,)
Pb
IV. PROPOSED EXPERIMENTAL PLAN
The procedure planned for proposed experimentation
is briefly discussed here. Single cylinder, four stroke
stationary C.I. engine and air induction system with
supercharger is to be selected for the experimental
task. In stationary engine operating parameters like
load factor can be easily controlled since other factors
air resistance, road resistance and vehicle weight are
absent. Also value of gravitational acceleration (g) is
assumed to be constant because of fixed engine
location.Throughout the complete trials, engine jacket
cooling water and speed is to be keep constant. Air
induction pressure is to be controlled for the analysis
of this factor on engine performance. Supercharger
arrangement is to be use as a test piece for air
induction pressure variations. The different parameters
are to be keep at the planned level for the different
engine conditions. Different number of load variations
and air induction pressure variation are to be observe.
Thus different numbers of fuel consumption rates as
response variable are going to obtain by keeping other
parameters constant each time. Error or the deviations
in test might be due to the lack of control in holding
the variables at their planned levels or simple lack of
precision in the measurements. To minimize the error,
each observation is planned to take when the engine
setup will reaches at steady state condition [8-14].
V. DEVELOPMENT OF EXPERIMENTAL
DATA BASED MODEL
One independent Pi term (viz. ) and one dependent Pi
term (viz. ) have been identified in the design of
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
experimentation. These Pi terms are available for the
model formulation. Dependent Pi term is assumed to
be the function of the available independent Pi term.
The data of the independent and dependent parameters
of the system is to be gathered during the
experimentation. It is necessary to correlate
quantitatively independent and dependent Pi term
involved in this man-machine system. This correlation
is nothing but a mathematical model as a design tool
for engine system performance analysis. The optimum
values of the independent Pi terms can be further
decided for optimization of this model for maximum
efficiency with due considerations for the constraints
in each engine application.
5.1. Development of model for dependent Pi term
(Л1) :
From the analysis of Pi terms, Л1= f (Л2)
Where f stands for ―function of‖
A probable exact mathematical form for this
phenomenon could be
----------1.1
Equation (1.1) is one of the most important and
general relations assumed in scientific work and
common functional relationship obtained as a result
after applying dimensional analysis technique.
There are two unknown terms in the equation 1.1, viz.
constant of proportionality K1 and index a1. To get the
values of these unknowns minimum two sets of values
of Л2. As per the propesed experimental plan in design
of experimentation, different set of these values will
obtain. To compute the values of unknowns K1 and a1,
any arbitrary one set from data can be selected. Then it
may not result in one best unique solution representing
a best-fit unique curve for the remaining set of the
values. To be very specific to find out nCr
combinations of r sets taken together out of the
available set of the values. Solving these many sets
and finding their solutions will be a herculean task.
Hence to solve this problem by curve fitting technique
will be a good option. To follow curve fitting
technique it is necessary to have the equations in the
form as given below.
Z= a+b*x+c*y+d*z--------1.2
The equation 1.1 can be brought in the form of
equation 1.2, by taking the log of both sides of this
equation,
Log ( ) = log K1 + a1* log ( ) ----------1.3
Let, log ( ) = Z1, log K1 = K1‘, log ( ) = A,
Then the equation 1.3 can be written as
Z1= K1‘ + a1*A ------------- 1.4
Equation 1.4 is a regression equation of Z on A, an ndimensional co-ordinate system this represents a
regression hyper-plane.
= n K1‘ + a1*
*A= K1‘
+ a1*
*A -------- 1.5
Where n is the number of runs or the number of sets of
these values.
These equations are called normal equations
corresponding to the equation 1.4 and are obtained as
ISSN: 2231-5381
per the definition. In the above sets of equation values
of multipliers of K1‘and a1 are to be substituted for
computing values of unknowns (viz. K1‘and a1).
Values of terms on L.H.S. and the multipliers of
K1‘and a1 in the sets of equation 1.5 are to be
calculated. After substituting these values in the
equation 1.5, one set of equation to be solved
simultaneously, to get the values of K1‘and a1. These
equations can be solved using MATLAB to get the
values of
K1‘and a1.
VI. CONCLUSIONS
Dimensional analysis is best known as well as most
powerful tool to make a given test compact in
operating plan without losing generality or control. In
this work, reduction of variable is done using
dimensional analysis. After that procedure of proposed
experimental plan is explained this is necessary for
performing experiments and to extract expected result
of performance enhancement. These extracted results
can be used to formulate experimental data based
model. Unknowns can find out using MATLAB
software. The present work shows vital scope for
improvement of operating performance of C.I. engine
by optimization of the model formed.
ACKNOWLEDGMENT
Authors are thankful to the SSBT`s, College of
Engineering and Technology, Bambhori, Jalgaon for
providing library facility. The authors would like to
thank the staff and colleagues for useful discussions.
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Engine Performance Improvement" International Powertrains,
Fuels & Lubricants Meeting, Rio De Janeiro, Brazil, (2010)
SAE Paper No. 2010-01-1575.
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
[8] M.S. Deshmukh, D.S. Deshmukh and J.P. Modak.
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ISSN: 2231-5381
[11] D.S. Deshmukh, M.S. Deshmukh and A.A. Patil "Design
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