International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 603

ISSN 2229-5518

Modeling and Optimization of Performance of

Four Stroke Spark Ignition Injector Engine

Okafor A. A., Achebe C. H., Chukwuneke J. L., Ozoegwu C. G.

Abstract— The performance of an engine whose basic design parameters are known can be predicted with the assistance of simulation programs into the less time, cost and near value of actual. This paper presents a comprehensive mathematical model of the performance parameters of four stroke spark ignition engine. The essence of this research work is to develop a mathematical model for the analysis of engine performance parameter of four stroke spark ignition engine before embarking on full scale construction, this will ensure that only optimal parameters are in the design and development of an engine and also allow to check and develop the design of engine and it’s operation alternatives in an inexpensive way and less time, instead of using experimental method which requires costly research test beds. To achieve this, equations were derived which describe the performance parameters (sfc, thermal efficiency, mep, and A/F). the equations were used to simulate and optimize the engine performance of the model for various engine speeds. The optimal valves obtained for the developed bivariate mathematical models are: sfc is 0.2833kg/kwh, efficiency is 28.77% and a/f is 20.75.

Index Terms— Bivariate models, Engine performance, Injector engine, Optimization, Performance parameters, Simulation, Spark ignition

—————————— ——————————

1 INTRODUCTION

Oday simulation and optimization are important domains which attract many researchers from several fields and disciplines [1]. In recent years, extensive research has been conducted in the area of simulation to model large complex systems and understand their structures and behaviors. At the same time, a variety of design principles and approaches for
computer based simulation has evolved.
This paper will deal with the modeling and optimization of
performance of four stroke spark ignition engine. Performance
evaluation of automotive engines is of great importance for
their economic operation. The method or criteria for assessing
the engine performance include the determination of engine
power, engine efficiency, and fuel consumption, considering
the engine stroke, engine speed, mean effective pressure and
bore. All of these affect the horse power, engine efficiency and
its performance.
High engine efficiency means obtaining the greatest
possible power with lowest possible fuel cost or lowest fuel
consumption [2]. At the design and development stage an
engineer would design an engine with certain aims in his
mind. The aims may include the variables like indicated
power, brake power, brake specific fuel consumption, exhaust
emissions, cooling of engine, maintenance free operation etc.
The other task of the development engineer is to reduce the
cost and improve power output and reliability of an engine. In
trying to achieve these goals he has to try various design
concepts. After the design the parts of the engine are
manufactured for the dimensions and surface finish and may

————————————————

Okafor A. A. is currently pursuing doctorate degree program in mechanical engineering in Nnamdi Azikiwe University,Awka, Nigeria.

Achebe C. H. is currently an associate professor in mechanical engineering in Nnamdi Azikiwe University,Awka, Nigeria.

Chukwuneke J. L. is currently pursuing doctorate degree program in mechanical engineering in Nnamdi Azikiwe University,Awka, Nigeria.E- mail: jl.chukwuneke@unizik.edu.ng

Ozoegwu C. G. is currently pursuing doctorate degree program in

mechanical engineering in Nnamdi Azikiwe University,Awka, Nigeria.

be with certain tolerances. In order to verify the designed and developed engine one has to go for testing and performance evaluation of the engines.
Thus, in general, a development engineer will have to conduct a wide variety of engine tests starting from simple fuel and air-flow measurements to taking of complicated injector needle lift diagrams, swirl patterns and photographs of the burning process in the combustion chamber [3]. The nature and the type of the tests to be conducted depend upon various factors, some of which are: the degree of development of the particular design, the accuracy required, the funds available, the nature of the manufacturing company, and its design strategy.
Engine tests experiment require costly research engine test beds and skilful technicians to run them. In order to reduce the cost and time of engine design and development, it, therefore, becomes necessary to develop a computer simulation program using mathematical model that will describe and quantify engine performance process. This process is enhanced by using a computer to simulate and optimize engine performance before embarking on full scale construction. It will ensure that only optimal parameters are in the design and development of an engine. This would reduce the cost and time of engine design and development to a minimum.
This research work on the modeling, simulation and optimization of performance of four stroke spark ignition injector engine intends to develop a computer simulation program or models for simulating and optimizing engine performance before embarking on full scale construction, and to develop bivariate mathematical models and optimize the engine performance of the model various engine speeds. This research work is narrowed down to four stroke spark ignition injector engine. MATLAB toolbox library equation was used in the development of the simulation models. General non-linear multivariate least squares modeling of experimental result was carried resulting in specific bivariate models for various engine performance measures. This enabled analytical

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optimization of the engine performance using the tool of partial differentiation. Optimization is carried out on each of air-fuel ratio, SFC and thermal efficiency because the surface plots of their bivariate models revealed existence of curvature.

2 MATHEMATICAL MODEL OF THE SYSTEM

2.1 Theoretical Terms

Torque, T (Nm): The total effective load on the torque arm is the sum of the added weights plus the spring balance reading. Generally, the value of torque, T is the total weight applied multiplied by the torque arm length, L.

Torque, T = W+S L (1)

1000

Where; T = Torque (Nm), (W+S) = total weight applied,
Newton, L = torque arm Length or radius arm (m).
Power (kw): The power or rate of doing work is measured
responses must be minimum. The least squares approximation theory is chosen to be a criterion for mathematical modeling in this work.

2.3 Bivariate Quadratic Least Squares Model of the

System Responses

Generally speaking least squares approximation of a response from a set of scattered data is based on minimizing square of experimental error. The generalized presentation of non-linear least squares approximation theory that is found very applicable in bivariate analysis presented here is seen in [4].
Suppose provided are 𝑛 experimental responses 𝑧𝑖 at positions
𝒙𝑖 in a real d-dimensional parameter space, that is the vector
𝒙𝑖 𝜖𝑅𝑑 where 𝑖𝜖[1, 2, … . . 𝑛]. The least squares method seeks to
fix a function 𝑧(𝒙) that approximates the experimental
responses 𝑧𝑖 by minimizing the sum of squares of Euclidian
error norms‖𝑧(𝒙𝑖 ) − 𝑧𝑖 ‖. The error functional thus reads:
in watts (Nm/sec) or kilowatts and is defined as the torque
𝐸𝑁𝐿𝑆 = ∑𝑛
‖𝑧(𝒙𝑖 ) − 𝑧𝑖 2
multiplied by the angular velocity.
Power (p) = Tω (2)
Where; T = torque, ω = Average speed = (2πN)⁄60, N =
speed in rpm.
Fuel Consumption (kg/h): Having timed the
consumption of a control volume of fuel, the fuel
consumption in kg/h may be calculated as follows. For a
control volume of 100cc the consumption per second is
given by 100⁄t where t is the measured time.

𝑖 =1 (8)

The approximation polynomial is generally given in the form
𝑧(𝒙) = [𝒂(𝒙)]𝑇 𝒃 (9)
Where; 𝒂(𝒙) = {𝑎1 (𝒙) 𝑎2(𝒙) … … 𝑎𝑟 (𝒙)}𝑇 is the polynomial of
basis vector and 𝒃 = {𝑏1 𝑏2 … … 𝑏𝑟 }𝑇 is the vector of coefficients
of approximation function 𝑧(𝒙). For illustration the quadratic
bivariate approximation function 𝑧(𝒙) has dimension 𝑑 = 2
and order 𝑝 = 2 such that
𝑧(𝒙) = 𝑧(𝑥, 𝑦) = 𝑏1 + 𝑏2𝑥 + 𝑏3𝑦 + 𝑏4𝑥2 + 𝑏5 𝑥𝑦 + 𝑏6𝑦2 (10a)
Fuel consumption = V⁄t
(3)
𝑏1
Specific Fuel Consumption (kg/kwh): An important
⎧𝑏2
characteristic of an internal combustion engine is the specific fuel consumption which relates the thermal
𝑧(𝒙) = {1 𝑥 𝑦 𝑥2 𝑥𝑦 𝑦2 }
⎪ ⎪
𝑏3
𝑏
(10b)
efficiency of the engine. This is defined as the weight of
fuel required to generate each kilowatts hour of energy.
4
⎪𝑏5
⎩𝑏6
SfC = weight of fuel⁄power
(4)

Mass flow rate of fuel, MF : This is the quantity of fuel

consumed in kg/s or weight of fuel supplied in kg per
second.

(volume in litre x specific gravity)

From which it is seen that 𝒂(𝒙) = {1 𝑥 𝑦 𝑥2 𝑥𝑦 𝑦2 }𝑇 and
𝒃 = {𝑏1 𝑏2 𝑏3 𝑏4 𝑏5 𝑏6}𝑇 and 𝑟 = 6. As shown on the works [5-

6], the length or number of elements in 𝒂(𝒙) or 𝒃 is generally
MF =

(time in sec x 1000)


(5)
given as 𝑟 =

(𝑑+𝑝)!

𝑑!𝑝!

. At the experimental points Eq.(2) which
Efficiency, η: This is the ratio of brake power to energy
supplied by the fuel.
reads 𝑧(𝒙) = [𝒂(𝒙𝑖 )]𝑇 𝒃 is inserted in Eq.(1) to give:

𝑛 𝑇 2

η = BP/(Mf x Calorific value) (6)
𝐸𝑁𝐿𝑆 (𝒃) = ∑𝑖 =1‖[𝒂(𝒙𝑖 )] 𝒃 − 𝑧𝑖
(11)
Brake Mean Effective Pressure: Mean effective pressure
is defined as hypothetical pressure which is thought to be
acting on the piston throughout the power stroke. If it is
The coefficient vector 𝒃 that minimizes the error functional
𝐸𝑁𝐿𝑆 (𝒃) is determined by differentiating 𝐸𝑁𝐿𝑆 (𝒃) [7] with
respect to 𝒃 and equated to zero to give:
based on brake power it is called brake mean effective
𝑛 1 2𝒂(𝒙𝑖 ){[𝒂(𝒙𝑖 )]𝑇 𝒃 − 𝑧𝑖 } = 𝟎
(12a)

𝑛 𝑛

pressure.
This gives: ∑𝑖 =1 𝒂(𝒙𝑖 )[𝒂(𝒙𝑖 )]𝑇 𝒃 − ∑𝑖 =1 𝒂(𝒙𝑖 )𝑧𝑖 = 𝟎
(12b)



B. M. E. P. = bp = 60 ×1000 ×bp = 60000 ×BP
(7)
Eq. (5) is re-arranged to give the minimum-error coefficient

LAn

LAnK

LAnK

vector 𝒃 as: 𝒃 = {∑𝑛
𝒂(𝒙𝑖 )[𝒂(𝒙𝑖 )]𝑇 }−1 𝑛
𝒂(𝒙 )𝑧
(13)

𝑖 =1

𝑖 =1

𝑖 𝑖

2.2 Model Description

The laws governing real systems cannot result from pure analytical modeling because of complications of non-linearity.
The minimum-error coefficient vector 𝒃 as given in Eq.(13) is
inserted in Eq.(9) to give the needed approximation
polynomial as:
For this reason approximate system-specific models are
𝑧(𝒙) = [𝒂(𝒙)]𝑇 {∑𝑛
𝒂(𝒙𝑖 )[𝒂(𝒙𝑖 )]𝑇 −1 𝑛

𝑖=1 }

𝑖 =1 𝒂(𝒙𝑖 )𝑧𝑖
(14)
derived to obey experimentally determined system responses.
The approximate models are always determined on the
criterion that error between experimental and mathematical
For the quadratic bivariate approximation the function 𝑧(𝒙)
becomes:

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1
⎡ ⎧ 𝑥𝑖
⎢ 𝑦𝑖

𝑛

−1


2 2 𝑛

1
⎧ 𝑥𝑖
𝑦𝑖
𝑧(𝒙) = {1 𝑥 𝑦 𝑥2 𝑥𝑦 𝑦2 } ∑𝑖=1 2
⎢ ⎨ 𝑖
{1 𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑥𝑖 𝑦𝑖 𝑦𝑖 }
⎬ ⎥
𝑖=1 2
𝑖
𝑧𝑖

(14a)
⎢ 𝑥𝑖 𝑦𝑖

2

⎥ 𝑥𝑖 𝑦𝑖

2

⎣ ⎩ 𝑦𝑖
⎦ ⎩ 𝑦𝑖
This on multiplying out gives
⎡ ⎡ 1 𝑥𝑖 𝑦𝑖 𝑥𝑖
𝑥𝑖 𝑦𝑖 𝑦2

2 3 2 2

⎢ ⎢ 𝑥𝑖 𝑥𝑖
𝑥𝑖 𝑦𝑖 𝑥𝑖
𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖 ⎥⎥
⎧ 𝑥𝑖

2 2 2 3

𝑦𝑖
𝑛
⎢ 𝑦𝑖 𝑥𝑖 𝑦𝑖 𝑦𝑖
𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖
𝑦𝑖 ⎥⎥ 𝑛
𝑧(𝒙) = {1 𝑥 𝑦 𝑥2 𝑥𝑦 𝑦2 } ⎢∑𝑖=1 2 3 2

4 3 2

2 ⎥⎥

𝑖=1
𝑥2
𝑧𝑖

(15)

⎢ ⎢ 𝑥𝑖
𝑥𝑖
𝑥𝑖 𝑦𝑖 𝑥𝑖
𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖 ⎥⎥
⎨𝑥 𝑖

2 2 2 2 2 3

𝑖 𝑦𝑖

⎢ ⎢𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖
𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖
𝑥𝑖 𝑦𝑖 ⎥⎥
⎩ 𝑦2

2 2 3 2 2 3 4 𝑖

⎣ ⎣ 𝑦𝑖
𝑥𝑖 𝑦𝑖
𝑦𝑖
𝑥𝑖 𝑦𝑖
𝑥𝑖 𝑦𝑖
𝑦𝑖 ⎦⎦
Eq.(15) is used in the modeling and optimization of engine
performances as presented below. In the engine performance
test experiment, the rotational speed [rev/min] designated 𝑥
and torque [Nm] designated 𝑦 are the independent variables
while each of brake power [Kw], time for consumption of
100cc [s], air-fuel ratio, specific fuel consumption (SFC)
[Kg/kwh], thermal efficiency and mean effective pressure
(MEP) [bar] are the responses designated 𝑧(𝑥, 𝑦). In order to make use of Eq.(15) for each of the responses 𝑧(𝑥, 𝑦) a
summation table is formed as shown in table.1.

2.4 Methodoly for use of the System Model

The idea of general non-linear multivariate least squares regression is used in generating a novel bivariate approximation of the studied system as presented in Eq.(15) The general quadratic bivariate model has the form:
𝑧(𝑥, 𝑦) = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑦 + 𝑎3 𝑥2 + 𝑎4 𝑥𝑦 + 𝑎5 𝑥𝑦2 (16a)
This can be put in the vector form:
𝑎0
⎧𝑎1
⎪𝑎2
is then plotted against both 𝑥(𝑠𝑝𝑠𝑠𝑑) and y (torque) [on the
horizontal axis].
This is simply done by plotting Eq.(16) in which 𝒙 and y on
the horizontal plane constitute the independent variables and
z is the modeled dependent variable. The same thing is done
for all the other bivariate models.

TABLE 1

SUMMATION TABLE FOR THE EXPERIMENTAL COODINATES

𝑥
𝑥
𝑥 𝑦
𝑧(𝒙, 𝒚) = {1, 𝑥, 𝑦, 𝑥2 , 𝑥𝑦, 𝑦2 } 𝑎
(16b)
3
⎪𝑎4
⎩𝑎5
It must be noted that the right hand side of Eq.(15) with
subscripts i is numerically calculated from the experimental
values to give the numerical coefficients 𝑎0, 𝑎1, 𝑎2, 𝑎3, 𝑎4,𝑎5,𝑎6.

The values of table 1 are summation values calculated from

experimental data and are used in the subscripted part of Eq.(15)

to generate the model coefficients ( 𝑎0, 𝑎1 , 𝑎2, 𝑎3, 𝑎4,𝑎5,𝑎6).

The accuracy of table 1 is absolutely guaranteed. For example
the summation associated with 𝑥(𝑠𝑝𝑠𝑠𝑑) in table 1 is 21000
which is derived from experimental results by summing
elements of first column as follows 4500 + 4000 + 3500 +
3000 + 2500 + 2000 + 1500 = 21000
All the bivariate models are calculated from Eq.(15) by
replacing the subscripted quantities with experimental
summation values in table.1 leading to the generation of
relevant coefficients ( 𝑎0, 𝑎1 , 𝑎2, 𝑎3, 𝑎4,𝑎5,𝑎6 ) of the models. The
meaning of the surface plots (Figs.1a–1f ) are obvious from
looking at the axis of the plots. Considering the brake power
for example, the bivariate model resulting from Eq.(15) is
Eq.(16) in which 𝒙 designates speed, y designates torque and z
designates the brake power. The brake power (on vertical axis)
𝑥
Making use of Eq.(15) and table.1 the following bivariate models and surfaces for the engine responses are calculated.

3 RESULTS AND OPTIMIZATION

3.1 Bivariate Model for Brake Power

𝑧(𝑥, 𝑦) = −8.1458 + 1.4851 × 10−3 𝑥 + 0.17215𝑦 − 9.1691 ×
10−8 𝑥2 + 9.1845 × 10−5 𝑥𝑦 − 9.5256 × 10−4𝑦2 (17)

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Fig.1a: The Brake power approximation surface

3.2 Bivariate Model for Brake Time

Fig.1c: The approximation surface for a/f ratio

3.4 Bivariate Model for Brake SFC

−4

𝑧(𝑥, 𝑦) = 1073.7 − 0.15266𝑥 − 19.101𝑦 + 5.6658 × 10−6 𝑥2 +
𝑧(𝑥, 𝑦) = −3.2038 + 3.1904 × 10
𝑥 + 0.075294𝑦 + 4.4981 ×

1.3286 × 10−3𝑥𝑦 + 9.0557 × 10−2 𝑦2 (18)
10−9
𝑥2
− 4.4705 × 10−6
𝑥𝑦 − 3.8412 × 10−4
𝑦2
(20)

Fig.1b: The approximation surface for Time of consumption of 100cc

3.3 Bivariate Model for Air/Fuel Ratio

Fig.1d: The approximation surface for SFC

3.5 Bivariate Model for Thermal Efficiency

𝑧(𝑥, 𝑦) = 358.06 − 0.030482𝑥 − 7.0822𝑦 − 3.9673 × 10−7 𝑥2 +
𝑧(𝑥, 𝑦) = 42.487 + 0.021151𝑥 − 0.47443𝑦 − 3.0075 × 10−6 𝑥2
9.024 × 10−5 𝑥𝑦 − 1.7503 × 10−4 𝑦2 (19)
4.2305 × 10−4
𝑥𝑦 + 0.036024𝑦2

(21)

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for optimization being that linear approximation could have given an adequately accurate result. Judging from the behavior of the generated approximation surfaces the candidate for optimization are air-fuel ratio, SFC and thermal efficiency. Of particular interest is the thermal efficiency for which the optimization is done first. The generated bivariate model is:
𝑧(𝑥, 𝑦) = 358.06 − 0.030482𝑥 − 7.0822𝑦 − 3.9673 × 10−7 𝑥2 +
4.2305 × 10−4𝑥𝑦 + 0.036024𝑦2 (23)
Where it should be recalled that 𝑥 represents speed, 𝑦
represents torque and 𝑧(𝑥, 𝑦) represents thermal efficiency. It
is seen from Fig.1e. that a clear cut optimal exist at which the

𝜕

theory of partial differentiation gives

𝜕𝑥


𝑧(𝑥, 𝑦) = 0 and

Fig.1e: The approximation surface for Thermal Efficiency

3.6 Bivariate Model for MEP

𝜕 𝑧(𝑥, 𝑦) = 0. As seen in Fig.1e this point has lifted wings due

𝜕𝑦

to experimental error because (𝑥, 𝑦) should vary
simultaneously. This point should have depressed wings for it
to be a point of maximum efficiency. Optimality thus satisfies
the pair of equations

𝜕 𝑧(𝑥, 𝑦) = −0.030482 − 2 × 3.9673 × 10−7 𝑥 + 4.2305 ×

𝜕𝑥

𝑧(𝑥, 𝑦) = −15.399 + 2.4873 × 10−3 𝑥 + 0.71472𝑦 − 1.4104 ×
10−4
𝑦 = 0 (24a)
10−7 𝑥2 − 2.2260 × 10−5 𝑥𝑦 − 1.8034 × 10−3𝑦2 (22)

𝜕 𝑧(𝑥, 𝑦) = −7.0822 + 4.2305 × 10−4

𝜕𝑦

𝑥 + 2 ×

0.036024𝑦 = 0 (24b)
Eq.(24) is rearranged into a system of linear equations:
𝑷𝒙 = 𝒑 (25)
Where 𝑷 is a 2 by 2 symmetric matrix given as
�−2 × 3.9673 × 10−7 4.2305 × 10−4
4.2305 × 10 2 × 0.036024
0.030482
7.0822 �.
Matrix inversion gives the optimal point as follows:
𝒙 = 𝑷−1 𝒑 =
�−2 × 3.9673 × 10−7 4.2305 × 10−4
4.2305 × 10 2 × 0.036024
𝒙 = �3387.7rev/min
78.407Nm
0.030482
� 7.0822 �,
This is the point of maximum thermal efficiency. When this
optimal point is inserted into Eq.(21), the maximum thermal efficiency becomes 28.777 percent.
The model generated for SFC is:
𝑧(𝑥, 𝑦) = −3.2038 + 3.1904 × 10−4 𝑥 + 0.075294𝑦 + 4.4981 ×
10−9 𝑥2 − 4.4705 × 10−6 𝑥𝑦 − 3.8412 × 10−4 𝑦2 (26)
From Fig.1d it can be seen that optimality exists when

𝜕 𝑧(𝑥, 𝑦) = 3.1904 × 10−4 + 2 × 4.4981 × 10−9 𝑥 − 4.4705 ×

𝜕𝑥

10−6 𝑦 = 0 (27)

Fig.1f: The approximation surface for MEP

𝜕 𝑧(𝑥, 𝑦) = 0.075294 − 4.4705 × 10−6

𝜕𝑦

𝑥 − 2 × 3.8412 ×

3.7 Optimization

An optimization problem is one requiring the determination of the optimal (maximum or minimum) value of a given function, called the objective function [8-9]. It has been stated
10−4 𝑦 = 0 (28)
It is seen from Eqs.(27) and (28) that
𝑷 = �2 × 4.4981 × 10−9 −4.4705 × 10−6
−4.4705 × 10 −2 × 3.8412 × 10
𝑷 = �−3.1904 × 10−4
earlier in the existing text that the brake thermal efficiency for
spark ignition engine which this work examines is between 20
and 30 percent [10]. Eligibility of each of the system’s
−0.075294
point as follows:
𝒙 = 𝑷−1 𝒑 =
. Matrix inversion gives the optimal

−1

responses for optimization is based on the configuration of the approximate surface. It is seen from the surfaces for Brake power and mean effective pressure that there is no question
�2 × 4.4981 × 10−9 −4.4705 × 10−6
−4.4705 × 10−6 −2 × 3.8412 × 10−4
𝒙 = �3402 rev/min
78.212 Nm
�−3.1904 × 10−4
−0.075294 ,

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When this point is inserted into Eq.(20) optimal SFC=0.28336 results. This is the optimal value for SFC.
is between 20 and 30 percent [11]. The optimal value for the
𝑆𝑝𝑠𝑠𝑑
SFC is calculated as SFC=0.28336 at the poin �𝑇𝑇𝑟𝑇𝑇𝑠� =
The model for air-fuel ratio is seen from Eq.(19) to be:
3402 rev/min
�. Optimal air-fuel ratio of 20.75 computed for
𝑧(𝑥, 𝑦) = 42.487 + 0.021151𝑥 − 0.47443𝑦 − 3.0075 × 10−6 𝑥2
78.212 Nm
𝑆𝑝𝑠𝑠𝑑
2341.5 rev/min
9.024 × 10−5 𝑥𝑦 − 1.7503 × 10−4 𝑦2 (29)
the engine to occur at �𝑇𝑇𝑟𝑇𝑇𝑠� = �
78.31 Nm �. It must be
It is clearly seen from the air –fuel ratio surface of Fig.1c that
optimality that is approximately constant with torque exists at some value of speed. Thus the idea of optimization used here is to keep Torque fixed and differentiate with respect to speed to give

𝜕 𝑧(𝑥, 𝑦) = 0.021151 − 2 × 3.0075 × 10−6 𝑥 − 9.024 × 10−5 𝑦 =

𝜕𝑥

0 (30)

−5


This is rearranged to give: 𝑥 = 0.021151−9.024×10 𝑦

2×3.0075×10

(31)
A convenient value for 𝑦 is then chosen to give an optimal
location for air-fuel ratio. Since each of thermal efficiency and
SFC has clear cut optimal locations that are very close to each
other it is recommended that 𝑦 value to be inserted in Eq.(31)
can be located at either the arithmetic or geometric mean of
the optimal locations of thermal efficiency and SFC. The
arithmetic or geometric means of the optimal locations of thermal efficiency and SFC are respectively given by
pointed out that the results of the generated bivariate models
for various engine performance correlate closely with those of
both experiment and MATLAB simulation. Based on the achievements or contributions listed above, it can be concluded that this research work has attained its set objectives.

REFERENCES

[1] J. Ruth, “Computer simulation Analysis of Biological and Agricultural Systems”, McGraw-Hill book company, New York, Pp.85, 2003.

[2] H. Akkinison., “Mechanics of small Engines”, Third Edition, Toronto

McGraw Hill, Corporation. Pp.85-100, 1981.

[3] J. B. Heywood, “ICE Fundamentals”, Mc Graw Hill Book Co., USA.

Pp.123-141, 1988.

[4] C. G. Ozoegwu , “Least squares approximated stability boundaries of milling process”, International Journal of Machine tools and Manufacture, 79 Pp.24-30, 2014.

1 3387.7rev/min
3402 rev/min

[5] D. Levin, “The approximation power of moving least-Squares

𝒙𝑎𝑚 = 2 ��
78.407Nm � + �
78.212 Nm ��

Math.”, Comp. 67, Pp.224, 1517–1531, 1998.

𝒙𝑔𝑚 = ��3387.7rev/min
= �3394.8rev/min

78.31Nm
3402 rev/min
3394.8 rev/min

[6] T. P. Fries, H. G. Matthies, “Classification and overview of mesh free

methods”, Tech. Rep. TU Brunswick, Germany Nr. 2003-03, Pp.123-

125, 2003.

78.407Nm � �
78.212 Nm � = �
78.309 Nm �

[7] V. Ganasan, ”Internal Combustion Engines” Second Edition, Tata

It is seen that both arithmetic and geometric means of the
optimal locations of thermal efficiency and SFC are almost
identical. Then 𝑦 = 78.31 say is inserted into Eq.(31) to give
𝑥 = 2341.5. The optimal air-fuel ratio of 20.75 occurs at 𝒙 =
2341.5 rev/min
78.31 Nm �.

4 CONCLUSION

A bivariate mathematical model for the analysis of engine Performance parameters of four stroke spark ignition engine was developed. The general non-linear multivariate least square modeling was carried out resulting in specific bivariate models for various engine performance measures. This enabled analytical optimization of the engine performance using the tool of partial differentiation. Optimization is carried out on each of air-fuel ratio, SFC and thermal efficiency because the surface plots of their bivariate models revealed existence of curvature. Optimization yielded that maximum thermal efficiency exists at the point

McGraw–Hill Publishing Company Limited, New Delhi. Pp16-21, 23-

25, 27-34,310-317, 2006.

[8] K. A. Stroud, J. B. Dester, “Advanced Engineering Mathematics”, Fourth Edition, Palgrave Macmillan, New York. Pp.940-960, 2003.

[9] Second international conference on Engineering optimization, September 6-9, 2010, Lisbon, Portugal. Journal of Applied Physics.

105, 094904, 2009, Pp.365-367, 2010

[10] R.. K. Rajput, “Thermal Engineering”, Seventh Edition, Laximi

Publications (P) Limited. New Delhi. Pp.1048-1051, 1075-1081, 1125-

1140, 2009.

[11] W. J. Gajda, W. E. Billes, “Engineering Modelling and Computation”, Houghton Miffilin Company, Boston, Pp.186-189, 1978.

𝑆𝑝𝑠𝑠𝑑
𝑇𝑇𝑟𝑇𝑇𝑠� = �
3387.7rev/min
78.407Nm � as 28.777 percent. Accuracy and
reliability of this optimum must be noted being that it has
been stated earlier in the existing text that the brake thermal efficiency for spark ignition engine which this work examines

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