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CHAPTER 5 IDEAL MODELS OF ENGINE CYCLES

5.1 INTRODUCTION

5.2 IDEAL MODELS

5.3 THERMODYNAMIC RELATIONS

5.4 Cycle analysis with constant ideal gas

working fluid

5.1 INTRODUCTIONProcesses in ICE

intake compression combustion expansion Exhaust

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Pressure-volume diagram of firing spark-ignition engine, rc=8.4, 3500 rpmpi=0.4 atm, pe = 1 atm

combustion

exhaust

intake

expansion

compression

5.2 IDEAL MODELS ()

We consider a set of models which provide useful insights into performance and efficiency of engines. Each cycle describes the approximation made for the engine combustion process.

With models for each of these processes, a simulation of a complete engine cycle can be built up which can be analyzed to provide information on engine performance. Each cycle discussed in this chapter is a consecutive sequence of process through which we can follow the state of the working fluid as the engine executes a complete operating cycle

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P-V Diagram - Gas Engine

Bottom Dead Center

Start of compression stroke -volume above piston filled with fuel/air mixture

BDCTDC

Volume

Pre

ssure

P-V Diagram - Gas Engine

BDCTDCVolume

Pre

ssu

re

Piston travels up, fuel/air compressed and pressure rises

P-V Diagram - Gas Engine

Top Dead Center

End of compression stroke -volume in cylinder at a minimum

BDCTDC Volume

Pre

ssu

re

P-V Diagram - Gas Engine

Then have an instantaneous introduction of heat, which increases pressure again

BDCTDCVolume

Pre

ssu

re

COMBUSTION

P-V Diagram - Gas Engine

Power Stroke

Pressure forces piston down, creating torque on crank shaft

BDCTDCVolume

Pre

ssu

re

P-V Diagram - Gas Engine

Piston reaches bottom dead center again, exhaust valve opens, burned fuel/air expelled

BDCTDC Volume

Pre

ssu

re

Figure 5.1 Pressure-volume diagram of ideal cycles. Unthrottled operation: (a)constant-volume combustion; (b) constant-pressure combustion; (c) limited-pressurecombustion. (d) throttled constant-volume cycle; (e) supercharged constant-volume

cycle.

(unthrottled)

(throttled)

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+

(supercharged)

Fresh Charge Metering System

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Partially openedor throttled

Sets of assumptions which simplify each of these processes to a form convenient for analysis are given in table 5.1

Process AssumptionsCompression (1-2) 1. Adiabatic and reversible (hence isentropic)Combustion (2-3) 1. Adiabatic

2. Combustion occurs at(a) Constant volume(b) Constant pressure(c) Part at constant volume and part at constant pressure

(called limited pressure)3. Combustion is complete ( )

Expansion (3-4) 1. Adiabatic and reversible (hence isentropic)Exhaust (4-5-6) and Intake (6-7-1)

1. Adiabatic2. Valve events occur at top and bottom center3. No change in cylinder volume as pressure difference across

open valves drop to zero4. Inlet and exhaust pressure are constant5. Negligible velocity effects

Table 5.1 Ideal models of engine processes

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5.3 THERMODYNAMIC RELATIONS

Thermodynamic relations are very useful from which important operating parameters and properties (T, p, u, v, h) of working fluid can be calculated. The overalloperating parameters of greatest interest which can be determined from a thermodynamic analysis of the engine operating cycle are:

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(5.1)

indicated fuel conversion efficiency

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indicated mean effective pressure (imep)

(5.2)

where = work per cycle

= sum of compression stroke work (WC)

and expansion stroke work (WE)

(5.3)

1) Compression stroke (1 - 2)

(5.4)

Since the process is adiabatic and reversible

(5.5)

And the compression work is

(5.6)

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2) Combustion process (2 - 3 or 2 - 3a - 3b)

2.1 Constant-volume cycle

and (5.7 a, b)

2.2 Constant-pressure cycle

and (5.7 c, d)

2.3 Limited - pressure cycle (has two parts of assumed

processes)

and (5.7 e, f)

and (5.7 g, h)

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3) Expansion stroke (3 - 4)

3.1 Constant-volume cycle

and (5.8 a, b)

Expansion work

(5.9)

3.2 Constant-pressure cycle

and and (5.10 a, b, c)

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Expansion work

(5.11)

3.3 Limited - pressure cycle

and and (5.12 a, b, c)

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Expansion work

(5.13)

Indicated fuel conversion efficiency

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Constant volume cycle

Constant pressure cycle

Limited pressure cycle

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When the exhaust valve opens at point 4, cylinder pressure isabove the exhaust manifold pressure and a blowdown processoccurs. In the ideal exhaust process model, this blowdownoccurs with piston stationary at BC. During this blowdownprocess, the gas which remains inside cylinder expandsisentropically, and escapes from the cylinder. The residual mass

fraction at point 6 is given by

which is the residual gas at state 6.

(m4> m5)

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The mixture temperature at the end of the intake stroke and at the start of the compression stroke (point 1) can be determined using the first Law of the open system

Figure 5.2 Definition of system boundary for thermodynamic analysis of ideal cycle processes.

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Then m6u6 - m4u4 = peVd - mehe ;Vd = V4-V6

;

: average exhausted gas state (averaged specific enthalpy of exhaust gas)

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4 6

Application of the first Law between points 6 and 1 (Intake process)

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where = specific enthalpy of the inlet mixture

and p1 = pi ; pi = inlet pressure

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6 1

Vd mi

In the four-stroke engine cycle, work is done on the piston during the intake and the exhaust processes.

The work done by the cylinder gases on the piston during exhaust is

The work done by the cylinder gases on the piston during intake is

The network to the piston over intake and exhaust strokes, called pumping work, is

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The pumping mean effective pressure, (pmep)

: (throttled)

: (supercharged)

Wc,ig Gross indicated work per cycle

(work applied to piston during compression

and expansion strokes)

Wc,in Net indicated work per cycle (work applied to piston per an entire cycle)

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(All W/Vd)

Throttled

Supercharged

The net indicated fuel conversion efficiency is related to the gross indicated fuel conversion efficiency by

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5.4 Cycle analysis with constant ideal gas working fluid

- Working fluid in these ideal cycles is assumed to be an ideal gas.

- Cp and Cv are constant throughout the engine operating cycle.

5.4.1 Constant-volume cycle

compression work

(1)

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u = cvT

expansion work

(2)

Quantity can be related to the temperature rise during combustion:

(3)

Indicated fuel conversion efficiency can be expressed as

(4)

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Since 1-2 and 3-4 processes are isentropic betweenthe same volume

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Isentropic relation

rcSo

Replacing into (4), we obtain

Figure 5.2 Ideal gas constant-volume cycle fuel conversion efficiency as a function of compression ratio

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Dimensionless indicated mean effective pressure

- In terms of

= specific internal energy per unit mass of working fluid

m = mass of working fluid

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(5)

(6)

At state (1): p1V1 = mRT1

And by the definition of rc, rc

rc Vd

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Then using and

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(from R = Cp- Cv)

It is also useful to compare imep with the maximum

pressure in the cycle,

From the constant volume process (2-3)

From the isentropic process (1-2)

(7)36

(8)

Relation of can be obtained from eq. (3)

Thus, eq. (8) becomes

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(note that is known)

Mass fraction of residual gas xr at state 6 (end of exhaust)

: Isentropic expansion from state 4 to 5

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xr can be derived as

By similar analysis, T6 can be found by

(a)

Relation between T1 and temperature of fresh mixture induction Ti is

(b)

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Do u remember pe and pi ?!!pe = exhuast pressurepi = intake pressure

From (a) and (b), we obtain ratio of T1 and Ti

5.4.2 Limited and constant pressure cycle

Compression work

Expansion work

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CombustionConstant volume 2-3a

Constant pressure 3a-3b

Therefore the relation between amounts of heat converted from fuel and the increase in temperature,

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where

and

For constant volume (Otto) cycle

For constant pressure (Diesel) cycle

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Cutoff ratio is

Cutoff ratio is defined as a ratio of volumes after and before combustion :

For the same rc

Always greater than 1 !

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Thermal efficiency of Diesel cycle as a function of compression and cutoff ratios (k=1.4)

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Dimensionless mean effective pressure

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EXAMPLE 5.1 Many diesel engine can be approximated by a limited-pressure cycle. In a limited-pressure cycle, a fraction of the fuel is burnt at constant volume and the remaining fuel is burnt at constant pressure. Use this cycle approximation with = cp/cv = 1.3 to analyze the following problem:

Inlet conditions: p1= 1.0 bar, T1= 289 KCompression ratio: 15:1Heat added during combustion: 43,000 kJ/kg of fuelOverall fuel/air ratio: 0.045 kg fuel/kg air

Half of the fuel/air ratio is burnt at constant volume, then half at constant pressure.

Asked: draw a p-V diagram and compute the fuel conversion efficiency of the cycle.

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Analysis:

= 8314/29 = 287 J/(kgK)

;

=

= 957 J/(kgK)=

= 1244 J/(kgK)

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Limited pressure cycle (1-2-3a-3b-4)

compression process (1-2):

isentropic

Heat added during combustion

From the fuel/air ratio =

So heat added due to combustion

= 1852 kJ/kg mixture 48

Half of the total heat is burnt at constant volume and atconstant pressure = 1852/2 = 926 kJ/kg mixture

Constant volume combustion (2-3a):

Constant pressure combustion (3a-3b):

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Per 1 kg mixture

1 kg mixture

1 kg mixture

Then

Thus ANS

The fuel conversion efficiency calculated using equation (5.43)

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