This reciprocation engine incurring a significant loss between the piston and crankshaft ... Compression ratio is 8.36 (for ease of comparison with rotary engine).
Reciprocating Engine Efficiency Calculation Each piston having bore diameter= 2d unit which is also the diameter of the cylindrical casing. The crankshaft having a radius of d unit and length of connecting rod = l unit The connecting rod makes an angle ‘α’ with the plane of linear movement of the piston and the crankshaft making an angle
with the same plane wherein
ranges between
and
. The angle between the crankshaft and connecting rod is ( )= These above angles α and varies when the piston travels from BDC to TDC and vise versa. The volume of the combustion chamber at BDC is TDV and the volume of combustion chamber at TDC is BDV. These are the basic parameters for this engine design wherein all the angles are unitized by Radians and all temperature values are in absolute temperature Kelvin unless otherwise specially specified in this literature. The terms TDV stands for Top Dead Volume or maximum volume, BDV stands for Bottom Dead Volume or minimum volume or clearance volume.
Conditions: At TDC, (
At BDC,
)
(
In ideal condition for perfect operation, l>
and
) and
is the situation where
power transmission rate is maximum in an expansion cycle or compression cycle. For length travel by piston ́ (
)
(
) ………………………………………(1)
Loss factor calculation: This reciprocation engine incurring a significant loss between the piston and crankshaft when piston or crankshaft moves by the influence of gas pressure or by forward momentum respectively. This loss is factorized below as loss factor and is figured in Fig:2.
* (
(
)+ )………………………….(2)
Three different stages: For compression cycle, at Stage 1 Fig:3, the piston is in BDC. At Stage 2, when the piston in a position of maximum power transmission or when the stage 3, when the piston is in TDC.
and at
Similarly, for expansion cycle at Stage 1 Fig:2, the piston is in TDC and it is approaching toward stage 3 via stage 2 wherein at Stage 2, when the piston in a position of maximum power transmission or when . For both compression and expansion cycle, At stage 2, (
and )
Loss factor at different stages At Stage1 as
.
At Stage2 as (
).
At Stage3 as (
) .
Average loss factor calculation between adjacent stages: Average loss factor between stage 1and 2 (
̅̅̅̅̅ (
(
)
∫
)
,
∫
)
∫
(
(
(
)
))
-
(
*
∫
)
(
)
+
( )
Average loss factor between stage 2 and 3 ̅̅̅̅̅ (
(
)
(
*
∫
If,
∫ ̅̅̅̅̅
(
)
))
(
)-
)
)
(
(
)
(
(
*
(
)
*
∫
)
∫ (
(
,
∫
∫
)
+ +
+ ,
( )
and
, then
)
And, ̅̅̅̅̅
(
)
Average length travelled calculation between adjacent stages: Average length travelled between stage 1and 2 ̅̅̅̅̅̅ ́
(
)
(
)
(
)
(
)
(
∫
∫
∫
,(
)
)
)(
,(
∫
*(
)
)(
,(
( ((
)
)(
)+ )
((
(
))-
)
)
((
))
)-
……………………………………(5) ̅̅̅̅̅̅ ́
(
)
∫
∫(
)
*(
)
(
)+
(
)
(
)
∫
,(
)(
,(
)(
)
(( )
) (
(
)))-
……………………………………..(6)
Practical engine concept with standard ratios Practically a reciprocating engine is generally a ‘square’ engine to enhance efficiency where this square’ engine refers to a stroke to bore ratio be 1:1. Moreover the rod ratio is 1.75:1 for standard engine design and this rod ratio is the ratio between connecting rod and stroke length and it is designed generally ranges between 1.4 to 1.8.
Calculation of affected parameters of the engine , wherein
Compression ratio is 8.36 (for ease of comparison with rotary engine)
=> => (let us assume the unit is meter) Normal atmospheric condition
KPa= 101.3 KN/m2 From pv diagram m3 m3 =( )
=>T2 = 298 ( = 563.48 K
)
=( )
x ( )
=
= 101.3 x (
)
= 1601.32 KN/m2 So, T2 = 563.48 K = 1601.32 KN/m2 In combustion stage
Volume remains constant
=
and, T3 = T2 + .Q. wherein, fuel-air ratio =
, So fuel-mixture ratio, =
Q = Lower Heating Value (LHV) of gasoline = 42700 KJ/Kg R= 0.287 KJ/Kg.K KJ/Kg.K T3 = 563.48 +
=
=
x
=
x 42700 x
x 1601.32
= 9677.68 KN/m2
The three stages in Expansion cycle At Stage 1: = P3 = 9677.68 KN/m2 = 3405.43 K
= 3405.43 K
At Stage 2: (
,
)
(
(
))-
m3 *
+ *
+
= 9677.68 x (
)
= 1514.53 KN/m2 Now, =>
=(
)
=
x(
)
= 3405.43 x (
)
= 2219.71 K Average pressure from stage 1 to 2 As we all know pressure is the solely dependent on volume change in a combustion chamber and so the average pressure is dependent on the average volume. The average displacement of the piston from stage 1 to 2, yields an average displacement volume of the piston in the combustion chamber and the corresponding pressure to average displacement volume is the average pressure between stage 1 to ́ ́ 2. So from equn(5), ̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅̅ m3 ̅̅̅̅̅̅̅
̅̅̅̅̅̅
*̅̅̅̅̅̅̅+ *̅̅̅̅̅̅̅+ [ KN/m2
Average loss factor from stage 1 to 2
]
Average loss factor is a term which represents the resultant work output factor after the loss from piston to crankshaft via connecting rod. To calculate this factor from stage 1 to 2, the equation (2) becomes equn (3) and after solving the integral by putting value of ̅̅̅̅̅
, we get
i.e. 36% of the total work from piston at stage 1 to 2 is lost due to
the linkage between piston to connecting rod and connecting rod to crankshaft. This loss factor is the main ingredients for traditional reciprocating engine of being less efficient than the current innovation. Although traditional efficiency calculation based on only compression ratio may show a reciprocating engine efficiency of 60% but it does not accounting the loss in the linking among piston, connecting rod and crankshaft because the traditional calculation based on compression ratio is just only measuring the efficiency within the combustion chamber or measuring the efficiency before said linking mechanism and that’s why the traditional theoretical efficiency deviates much from the practical scenarios. So it is extremely vital for any new engine to account the loss in linking mechanism to compare the efficiency with the traditional one. From fig: it is clear that the average pressure, ̅̅̅̅̅̅ travels a distance of ,
-
At Stage 3: *
(
) m3= TDV
* *
So,
+
+
= 9677.68 x (
)
= 612.22 KN/m2 =(
Now, =>
=
) x(
)
(
( ))+
= 3405.43 x (
)
= 1801 K Average pressure from stage 2 to 3 The average displacement of the piston from stage 2 to 3, yields an average displacement volume of the piston in the combustion chamber and the corresponding pressure to average displacement volume is the average pressure between stage 2 to 3. In reciprocating engine the BDV or clearance volume does not transform, i.e. BDV or clearance volume is independent to the constraint by the stroke length, as BDV is not the derivative of (crankshaft angle with sliding plane of piston), i.e. . So the only transformation happened in the volume, precisely due to variation in the stroke length and that’s why average stroke length is calculated in order to get the average stroke volume as x sectional area, is uniform throughout the whole stroke length and that average volume will also indicate the corresponding pressure to be the average pressure within the projected and desired stage. So from equn(6), average stroke ́ length=̅̅̅̅̅̅ m ̅̅̅̅̅̅ ̅̅̅̅̅̅ ́
m3 ̅̅̅̅̅̅̅
̅̅̅̅̅̅
*̅̅̅̅̅̅̅+ *̅̅̅̅̅̅̅+ [
]
KN/m2 Average loss factor from stage 1 to 2 Average loss factor is a term which represents the resultant work output factor after the loss from piston to crankshaft via connecting rod. To calculate this factor from stage 2 to 3, the equation (2) becomes equn (4) and after solving the integral while putting value of
, we get,
̅̅̅̅̅
i.e. 40% of the total work from piston at stage 2 to 3 is lost due to the linkage among piston, connecting rod and crankshaft.
From fig: it is clear that the average pressure, ̅̅̅̅̅̅ travels at a distance of ,
-
Gross expansion work done ̅̅̅̅̅
̅̅̅̅̅̅ ̅̅̅̅̅
̅̅̅̅̅̅
KJ If an observation can be done in above equation, we can see that maximum work is done on stage 1 to 2.
The three stages in Compression cycle At Stage 1: = P3 = 101.3 KN/m
2
= 298 K m3 At Stage 2: The value of volume, for compression cycle at stage 2 when the piston moves on the way from TDV to BDV, is also the same value of the of the expansion cycle. Only difference between the volumes of expansion and compression cycle is that the of expansion cycle increased from BDV while the of compression cycle is decreased from TDV.
m3 *
and
+ *
= 101.3 x (
+
)
= 250.62 KN/m2 =(
Now, =>
) x(
=
)
= 298 x (
)
= 367.28 K Average pressure from stage 1 to 2 The average displacement of the piston from stage 1 to 2, yields an average displacement volume of the piston in the combustion chamber and the corresponding pressure to average displacement volume is the average pressure between stage 1 to 2 and the average stroke length at stage 1 to 2 in compression cycle is same as the average stroke length at stage 2 to 3 in expansion cycle. So from ́ equn(6), ̅̅̅̅̅̅ and average volume in combustion chamber is ̅̅̅̅̅̅. ̅̅̅̅̅̅ ̅̅̅̅̅̅̅
̅̅̅̅̅̅ ́
m3
*̅̅̅̅̅̅̅+ ̅̅̅̅̅̅
*̅̅̅̅̅̅̅+ [ KN/m2
Average loss factor from stage 1 to 2
]
Average loss factor is a term which represents the resultant work output factor after the loss from piston to crankshaft via connecting rod. To calculate this factor from stage 1 to 2, the equation (2) becomes equn (4) and after solving the integral by applying value of ̅̅̅̅̅
, we get,
i.e. 40% of the total work from piston at stage 1 to 2 is lost due to
the linkage between piston to connecting rod and connecting rod to crankshaft and the loss factor in compression cycle from stage 1 to 2 is same as the loss factor in expansion cycle from stage 2 to 3. From fig: it is clear that the average pressure, ̅̅̅̅̅̅ travels at a distance of the distance travelled by average pressure in expansion cycle from stage 2 to 3. ,
which is same as
-
At Stage 3: *
(
)
(
( ))+
m3 *
+ *
+
= 101.3 x (
)
= 1601.29 KN/m2 Average pressure from stage 2 to 3 The average displacement of the piston from stage 2 to 3, yields an average displacement volume of the piston in the combustion chamber and the corresponding pressure to average displacement volume is the average pressure between stage 2 to 3. So from equn(5), ̅̅̅̅̅̅ ́ ́ ̅̅̅̅̅̅ ̅̅̅̅̅̅ m3 ̅̅̅̅̅̅̅
̅̅̅̅̅̅
*̅̅̅̅̅̅̅+ *̅̅̅̅̅̅̅+ [
]
KN/m2 Average loss factor from stage 1 to 2 Average loss factor is a term which represents the resultant work output factor after the loss from piston to crankshaft via connecting rod. To calculate this factor from stage 2 to 3, the equation (2) becomes equn (4) and after solving the integral, we get, ̅̅̅̅̅
i.e. 36% of the total work from piston at stage 2 to 3 is lost due to the linkage among piston, connecting rod and crankshaft and the loss factor in compression cycle from stage 2 to 3 is same as the loss factor in expansion cycle from stage 1 to 2. From fig: it is clear that the average pressure, ̅̅̅̅̅̅ travels at a distance of and distance travelled during compression cycle in stage 2 to 3 is same as the distance travelled in expansion cycle from stage 1 to 2. ,
-
m3
Gross compression work to be done ̅̅̅̅̅̅
̅̅̅̅
̅̅̅̅̅̅
̅̅̅̅
KJ KJ If an observation can be done in above equation, we can see that maximum work to be done by the crankshaft is on during stage 2 to 3. It is also noticeable on above equation that the loss factor is multiplied to the pressure exerted in expansion cycle but the exerted pressure is divided by the loss factor in compression cycle. The reason behind that in expansion cycle, work is done to the crankshaft after the loss but in compression cycle the work is required to be done by the crankshaft to make the required changes in volume of combustion chamber. Now, Efficiency, We know,
,
-
Kg
KJ
(
)
The above calculations are based on ideal Otto cycle. However in reality the pv diagram does not follow the ideal Otto cycle 100% and in reality the above calculated efficiency is further decreased to 15-20% for set compression ratio of 8.36. The reason behind the decreased efficiency is that among the above calculated efficiency, a fraction of power is also lost to drive camshaft, to drive gear train and also for the uncontrollable heat transfer or loss via cylinder walls. Although engine efficiency is increased if the compression ratio is increased according to traditional calculation for efficiency but this compression ratio cannot increased without limit because compression ratio more than 12 is likely to produce a tremendous temperature in combustion chamber and subsequently increases the overall costing of the vehicle or power generation system.
