INTRODUCTION
The basic function of the engine intake pipe is to provide the engine with
a fresh charge for combustion in cylinders to take place every cycle. It is
therefore important to do a proper intake pipe design. The design of the intake
pipe represents a key area in which the automobile engineer can significantly
influence engines performance. For both spark and compression ignition engines,
the fluid flow processes with intake pipe play an essential role in determining
an engine’s overall operating characteristics. The power output of an internal
combustion engine is directly proportional to the amount of the charge (air
or mixture) that can be forced into the cylinder per cycle. This amount is most
effectively increased by means of a mechanical supercharger or a turbocharger.
These devices have several disadvantages. First, they add to the mechanical
friction or pumping work of the engine and second they raise the inlet charge
temperature of the engine, increasing the sensitivity of the engine to knock.
However, the power output may also be increased by the selection of a suitable
diameter and length for the intake pipe. Many researchers have found that the
amount of charge forced into the cylinder can be increased almost 1520% by
this method (Heywood, 2002).
The intake pipe is considered as a device for storing kinetic energy. It is
argued that as the charge flows toward the cylinder, the kinetic energy is stored
in the charge column by virtue of its velocity. Some of this energy can be recovered
toward the end of the inlet process because of the ramming effect of the charge
column which tends to build up the pressure at the intake port and forces more
charge into the cylinder. The length of the intake pipe should provide a frequency
(oscillation) of flow process of charge column movement conformity with the
engine, while, the diameter of the intake pipe should provide a sufficient store
in kinetic energy for this charge column at a minimum losses of energy that
required to overcome various types of aerodynamic resistances (the air filter,
major and minor losses, losses in valves mechanisms) (Heywood,
2002; Taylor et al., 1955). From this point
of view, it is important therefore to design an intake pipe geometry (its length
and diameter) that will reduce the impact of this flow restriction.
Research has focused (either experimentally or numerically) on the effect of
flow behavior in the intake pipes which can be used to force additional air
(or mixture) into the engine making it more efficient. The level of the influence
is mainly determined by the pipe diameter and length. Until recently, the design
of such problem has been based on a trial and error which is costly at time
consuming. Early intake pipes were based on any commercially available pipes
without regarding the behavior to interior flow as shown in Fig.
1 (Eberhard, 1971). Many theories were built forward
regarding the behavior of the flow in the intake pipe. The numerical techniques
and methods (such as method of characteristics, LaxWendroff method, finite
element method, etc.,) were used to determine the optimum intake pipe length
and diameter to achieve improved volumetric efficiency. A number of papers have
been published on this subject, from 1D, 2D to 3D like (Ferguson
and Kirkpatrick, 2001). In addition, there are a number of commercial codes
available today (like KIVA, LOTUS) (Pearson et al.,
2006).

Fig. 2: 
Measured volumetric efficiency versus engine speed and intake
pipe length (Tabaczynski, 1982) 
Tabaczynski (1982) showed experimentally the effect
of intake pipe length on the volumetric efficiency with different engine speeds
as shown in Fig. 2. Analytical model was suggested to predict
the effective length of the intake pipe (Ferguson and Kirkpatrick,
2001; Tabaczynski, 1982).
The main object of this study is concerned with the mathematical description of such intake pipe diameter. An analytical solution to estimate the effective diameter of a naturally aspirated internal combustion engine intake pipe is presented. A formula is created to predict the effective diameter for the geometry of the intake pipe as a function of cylinder bore, stroke, engine speed, pipe length and aerodynamic losses. A comparison between the results of the novel suggested formula and the actual published work to determine the intake pipe diameter is done for different engines.
MATHEMATICAL ANALYSIS
The work can be divided into the following main steps. Firstly, is to find an algebraic expression for the kinetic energy of a charge column in the intake pipe during the period of filling the cylinder. This analysis is taken into considering the pressure loss that required to overcome the major and minor resistances. Secondly, is to obtain the maximum kinetic energy by differentiating the equation obtained in the first step relative to the intake pipe diameter then equating to zero. The resulting solution of this equation leads to predicate the best formula to design the diameter of the intake pipe. Finally, the diameter estimation of this form is validated with the actual model.

Fig. 3: 
Representation of intake pipe for four stroke aspirated internal
combustion engine (Tabaczynski, 1982) 
The approach for providing the maximum charge flow to engines as shown in Fig.
3, has to build intake pipes with large diameter which yield small charge
losses due to friction owing to small charge velocities in the pipes. The opposite
situation that of smaller diameter and higher velocities may also aid the charge
flow to the engine, due to use of the kinetic energy of the charge for the purpose
of “ramcharging” the cylinder, to compress the cylinder charge prior
to intake valve closure. The “tuned” pipe makes use this phenomenon
to increase the pressure in the cylinder at valve closure, thereby increasing
the mass of charge taken in by the cylinder (Heywood, 2002;
Ferguson and Kirkpatrick, 2001).
The intake pipe l_{c} in which all local resistances are included by increasing the origin intake pipe length l_{S} by an additional virtual length pipe l_{eq}.
Hence:
The total pressure loss of the pipe is the same as that produced in a straight pipe whose length is equal to the pipe of the original system plus the sum of the equivalent length of all the components in the system.
Using the famous formula for DarcyWeisbach, the pressure loss considering
the major and minor losses can be calculated:

Fig. 4: 
Representation model solution of the corrected intake pipe 
From that, the equivalent length is:
Thus, the new intake pipe length is the greater length l_{c} possesses aerodynamic properties of the real intake pipe length l_{p}, however, this kind is more convenient for the mathematical solution of this problem.
According to Fig. 4, the kinetic energy per unit volume in the outlet of pipe:
Moreover, from continuity equation can be obtained:
where, ξ is a coefficient of corrected velocity factor.
Then, substituting Eq. 5 in Eq. 4 results:
The law of movement for one dimensional in the differential form (Benson,
1982) is:
It seen that:
Alternatively, in other form:
The work done per unit mass of a gas in the pipe can be calculated as in the following:
The total work done by the gas with considering its velocity (Estop
and McConkey, 2009):
In a differential form:
Here, dW = 0, that is:
And, as:
Now from the law of conservation of mass (M_{S} = ρ_{S} u_{S} f):
And, hence:
Thus:
In addition, that at steady flow:
From that, can be obtained:
Substituting the Eq. 8, 10, 11
and 12 in the Eq. 7 yields:
The replacement of P_{S} by an equivalent value in the equation of state of gas and making transformations and simplifying, gives:
Now, equating equation of state for gases with Eq. 10:
After integration, results:
Substituting by equivalent value in Eq. 11:
Substituting value RT_{S} in Eq. 13, after transformations, becomes:
Integrating this equation along the pipe length results in:
In Eq. 15 the first term can be neglected because it is small. Also, assume that the accepted limit velocity of flow is about 40 m sec^{1}, so, the first part in the second term is small comparing with the second part so that can be neglected. Hence, from the last equation, we can get an expression for the density as follow:
So, the expression for the kinetic energy (Eq. 6) becomes:
The maximum kinetic energy can be obtained by differentiating the above equation relative to the pipe diameter and equating the resulted equation to zero.
After equating to zero and some transformations, Eq. 18 becomes:
The average speed of the piston is:
(Nrevolution per minute of the engine), the following equation is obtained:
The variation in the temperature in the intake pipe is closely equal to the ambient temperature. Also, substituting in last formula the following:
and R = 287 J/kg.K results:
From the experimental data, ξ_{0} = 1.48 in aspirated internal
combustion engines can be used (Ferguson and Kirkpatrick,
2001):
In addition, for highspeed engines especially, it is possible to set and accept the equivalent length simply equal to 1214 m and λ = 0.018 practically. Thus with the aim of this correction factors, the following expression can be obtained:
Therefore, the calculations of d_{S} can be carried out corresponding
to a rough estimate of the effective length l_{S} under the formula
given by Tabaczynski (1982):
where, a is a speed of sound: .
Equation 23 is the required relationship to estimate the
diameter of the inlet pipe as a function of cylinder bore, stroke, engine speed
and pipe length when considering the flow losses.
VERIFICATION OF RESULT
Verification is best achieved by means of comparison with the actual values
for the intake diameter of an aspirated internal combustion engines. The following
data was obtained from the published work. Many applications to examine the
resulting equation were done and some of these as in the following cases:
Case 1: 
The first application of obtained equation on the Dorman engine
(Data from Benson and Baruah work) (Benson, 1982),
with the following specifications: D = 125 mm, S = 130 mm, N = 1500 rpm,
l_{p} = 609 mm 
In Benson and Baruah work, the diameter is 50.8 mm.
Case 2: 
A Volkswagen engine with the following specifications: D =
82 mm, S = 110 mm, N = 1500 rpm 
On other hand, the actual diameter of intake pipe is about d_{S}≈
3336 mm.
Case 3: 
Another application of the obtained equation on the engine
of automobile BMW, with the following specifications: D = 80 mm, S = 6.6
mm, N = 6000 rpm 
Actually: d_{S} = 46 mm
Clearly, it is noted that the difference between the value of the intake pipe diameter which is calculated from the above formula and the actual value, is small.
CONCLUSION
The theoretical scheme described in this paper could successfully predict the intake pipe diameter of a naturally aspirated internal combustion engines.
Algebraic expression for the diameter of the intake pipe in the naturally aspirated
internal combustion engines has been obtained as a function of cylinder bore,
stroke, engine speed, ambient temperature, pipe length and losses.
The design equation shows a good correspondence with the values obtained from
the actual engines. Clearly, depending on the obtained calculation, the technique
offers a choice of an effective diameter of the intake pipe to provide essentially
a desirable result, especially in aspirated highspeed piston engines by inertia
supercharging as compared with previous results (Benson, 1982;
Eberhard, 1971) and that can be recommended for using
in the engineering practical field.
Obviously, the equation capable of accurately predicting the intake pipe diameter would be a powerful tool for the engine designer.
NOMENCLATURE
a 
: 
Speed of sound 
V 
: 
Mean piston speed 
D 
: 
Diameter of the cylinder 
d 
: 
Pipe diameter 
F 
: 
Crosssection area of the cylinder 
f 
: 
Crosssection area of the pipe 
k 
: 
Ratio of specific heats 
l 
: 
Pipe length 
N 
: 
Engine speed 
P 
: 
Pressure 
u 
: 
Velocity 
R 
: 
Gas constant 
S 
: 
Stroke 
T 
: 
Temperature 
Greek letters:
ξ 
: 
Correction factor of velocity 
λ 
: 
Coefficient of friction losses (major losses) 
μ 
: 
Coefficients of local aerodynamic losses (minor losses) 
ρ 
: 
Density 
υ 
: 
Specific volume 
Subscripts:
0 
: 
Ambient conditions 
1 
: 
Upstream conditions 
2 
: 
Downstream conditions 
i 
: 
Inlet 
b 
: 
:Bend 
c 
: 
Corrected 
eq 
: 
Equivalent 
p 
: 
Pipe 
v 
: 
Valve 
s 

Current conditions 