Xin Q. Diesel Engine System Design

176 Diesel engine system design

is very low, condensate of highly corrosive compounds may form, especially

sulfuric acid. The SO

3

reacts with the water vapor to form sulfuric acid

(H

2

SO

4

). The presence of acid increases the corrosion at lower component

temperatures close to the acid dew points. The sulfuric acid in the exhaust

gas will condense at near 150∞C; and the dew point margin is believed to

be the most important variable that affects corrosion (Kass et al., 2005).

The corrosion rate is dependent on the condensation and formation of

sulfuric acid, which is a function of fuel sulfur level, EGR rate, dew point

temperature margin and ambient humidity. The diesel fuels with high sulfur

level (e.g., 350 ppm sulfur) combined with condensation at the EGR cooler

outlet produce high corrosion rates in the engine components, while the

ultra-low-sulfur diesel fuel (with less than 15 ppm sulfur) does not produce

signicant corrosion (Kass et al., 2005). Sulfuric corrosion can be minimized

by using the ultra-low-sulfur fuel and by limiting the condensation of water

to avoid acid condensation in the EGR circuit. Sulfuric acid condensation

in EGR cooler was discussed by McKinley (1997), Kreso et al. (1998) and

Mosburger et al. (2008). Acidic condensation for diesel and biodiesel fuels

was investigated by Moroz et al. (2009).

Water condensate

There are two types of condensate in the engine: acid and water. Condensate

may cause three durability problems: in-cylinder lubricant oil dilution by

water, acidic corrosion, and aggravation of cooler fouling. Water condensation

may occur at any cooler outlet (e.g., charge air cooler, EGR cooler, inter-

stage cooler) and eventually in the intake manifold. EGR cooler condensation

occurs when the EGR cooler outlet gas temperature is lower than the local

condensation temperature. The condensation temperature is affected by the

gas pressure and humidity. At higher ambient humidity (higher dew point

temperature of water), higher EGR gas pressure or colder EGR cooler outlet

gas temperature, it is easier for the water in the EGR gas ow to condense.

The condensate rate is equal to the difference between the mass ow rate of

the H

2

O in EGR and the mass ow rate of saturated water vapor. The H

2

O

ow rate in EGR can be calculated based on the EGR rate and the H

2

O ow

in the exhaust gas. The H

2

O in the exhaust comes from the water contained in

the intake ambient air and the water formed during the combustion process.

The saturated vapor ow rate is calculated based on the saturation pressure

of the water vapor, which in turn varies with the gas temperature.

For heavy-duty engines, high water condensate rate at the EGR cooler

outlet usually occurs in regions of low speed and high load in the engine

speed–torque domain (e.g., around peak torque) due to the combination of

a relatively high EGR gas ow rate, high EGR gas pressure, and cold EGR

cooler outlet gas temperature. At rated power or low load, the water condensate

Diesel-Xin-02.indd 176 5/5/11 11:44:55 AM

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177Durability and reliability in diesel engine system design

rate is usually lower. The acceptable level of condensate is determined by

engine durability testing.

The sulfuric acid vapor condenses at a higher temperature than the water

vapor does. EGR gas contains acidic combustion products. Therefore, the

EGR water condensate is acidic and can cause corrosion problems. The gas

temperature at the EGR cooler outlet and the intake manifold needs to be

set sufciently high for a given fuel sulfur level in order to minimize the

durability risk due to water and sulfuric acid condensate. The EGR cooler

capacity is nally a design balance between the durability requirement of

water/acidic condensation and the emissions requirement of NO

x

control.

In summary, appropriate design targets should be selected and matched

properly in EGR cooler design, including carbon soot and HC ow rate,

gas–coolant temperature gradient, cooler outlet gas temperature, and condensate

rate.

2.13 Diesel engine system reliability

2.13.1 Objectives of engine system reliability analysis

Reliability has a great impact on brand image. Good reliability can reduce

product warranty costs. The objectives of engine system reliability analysis

include the following:

1. use the concept of design for reliability to ensure durability constraints are

properly selected in the early stage of diesel engine system design;

2. examine the reliability and lifetime of each subsystem or component,

and conduct reliability allocation to optimize the system lifetime and

cost.

For the rst objective, the concept of design for reliability requires a system

reliability target to be specied for a given period of mission time for the

population of the engine product to cover the variability. The probabilistic

nature of the reliability problems requires a nondeterministic approach to

be used in the analysis. System durability models are required in order to

estimate the lifetime of the engine or estimate the reliability for the statistical

population of the engine at a specied lifetime.

Depending on the maturity of the durability model for each component,

the second objective can be realized in either the early stage of system

design or a later stage of detailed design via system integration when the

component prototypes are available. A system reliability model is needed

as a function of component reliability. Reliability allocation needs to be

conducted between the components in order to meet the system reliability

target while satisfying the minimum requirement on the reliability target

for each component. Reliability importance and the impact of reliability

Diesel-Xin-02.indd 177 5/5/11 11:44:55 AM



178 Diesel engine system design

improvement on cost need to be evaluated. Nonlinear constrained optimization

can be conducted to minimize system cost. If some components are over-

designed for lifetime or reliability due to a great mismatch compared with

other on-target design components, their lifetime or reliability can be

reduced in order to save cost and reach a balanced lifetime between the

components. The following sections discuss the key concepts involved in

these processes.

2.13.2 Reliability engineering in diesel engine system

design

Reliability engineering is a well-developed discipline closely related to statistics

and probability theory. There are many areas in reliability engineering, for

example: reliability data analysis with the time-domain probabilistic models of

reliability, failure rate, and hazard rate by using time as the random variable to

address the probability of failure as a function of mission time (e.g., analysis

with the Weibull distribution); the stress–strength probabilistic interference

model by using design parameters as the random variable; reliability networks

of series, parallel, series–parallel and standby systems; FMEA and fault tree

analysis; and reliability-based optimization and design. Note that structural

durability failure mechanisms are usually not a major subject in reliability

engineering. When the principles of reliability engineering are applied to

diesel engine system design, the focus is to link durability with reliability and

develop probabilistic models of durability life estimation for major engine

components. Figure 2.9 illustrates the methodology of design for reliability

in engine system design.

The role of reliability in the design process was introduced by Dhillon

(1999, 2005), Kumar et al. (2006), Dietrich (2008), and Chandrupatla

(2009). The probability analysis in reliability engineering was elaborated

by Kapur (1992). Kuehnel et al. (2005) described a durability validation

process in reliability management for cooler development. Zhou and Li

(2009) summarized the key concepts in reliability validation processes in

engineering design.

Durability and reliability are two different concepts as discussed in Section

2.3. However, the concepts of durability and reliability become consistent

when the probability of failure is used. For example, a durability specication

of B10 life at one million miles (or equivalent number of engine hours)

represents that 10% of the engine population will fail within one million

miles. The equivalent reliability specication can be stated as the reliability

is 90% or the probability of failure is 10% at one million miles. An engine

system reliability model can be built using the probability distribution

equation with time as the random variable. The durability or reliability life

can be calculated by solving this equation for time, provided the failure

Diesel-Xin-02.indd 178 5/5/11 11:44:56 AM



179Durability and reliability in diesel engine system design

rate or reliability is a known input. It is desirable to include a reliability

requirement in the engine system design specication so that reliability is

designed-in at the beginning of product development.

2.13.3 Testing methods for reliability assessment

The reliability of a component, subsystem and system can be validated by

the methods of ‘pass/fail testing’ and ‘test to failure’ (Baker and Brunson,

2000). The pass/fail testing method is also known as ‘test to bogey’. Derived

from the binomial distribution model, the probability of no failure within a

specied test time duration is given by

P

pass

= R

n

= 1 – C

cl

2.12

where R is reliability, n is the sample size, and C

cl

is a condence level which

Steady-state durability design constraints of loading

Engine system design

Predicted loading levels (mechanical and

thermal, etc.) and loading cycles

The most severe condition in durability

validation and customer usage proﬁles/duty

cycles mapped in engine speed load domain

(full load steady state and transient time

history)

Material, heat transfer

and design

Durability test validation

Predict cumulative failure rate, durability and reliability

Damage indicator calculation for all failure modes

and components based on loading history

Construct statistical ‘stress–strength’ PDF curves with

Monte Carlo simulation for the entire population

Iterate

until reach

reliability

target

Damage indicator

Probability density

Stress

Strength

2.9 Methodology of design for reliability in engine system design.

Diesel-Xin-02.indd 179 5/5/11 11:44:56 AM



180 Diesel engine system design

is equal to the probability of one or more failures (Ireson et al., 1996). The

sample size required to prove the reliability level is then given by

n

C

R

=

ln

(

1 –

)

ln

cl

2.13

With the pass/fail testing method the failure distribution and the system life

at the end of test time are unknown. Moreover, the sample size required is

usually rather large. In contrast, although it takes longer to test, the method

of ‘test to failure’ can use a much smaller sample size (typically reduced

by in excess of 50% according to Baker and Brunson, 2000) and offer the

information of failure rate as a function of test time so that the probabilistic

distributions of the reliability of different systems can be analyzed and

compared. For example, the failure rate data can be  tted with a Weibull

distribution model.

The ‘test to failure’ method is destructive in order to obtain the failure

rate information. The test is expensive and often still requires a large number

of samples in order to gain statistical con dence. The smaller the sample

size, the higher the uncertainty. Probabilistic simulation may assist the

evaluation.

2.13.4 Probabilistic design for reliability

Time-based and stress- or damage-based probabilistic models

The time-based reliability model is shown below where the Weibull statistical

distribution is used as an example (see Table A.2(f) in the Appendix):

R(t) = e

–(t/b)

g

2.14

where time t is the random variable, b is the scale parameter and g is the

shape parameter. Note that this model does not contain the detailed design

information of why the failure occurs. Such a model can be obtained by

using a statistical distribution to  t the reliability warranty/service data, lab

test data, or simulation data. For high mileage engines, warranty/service

data collection can be very time consuming. As conducted by Daubercies et

al. (2009), repurchasing customer vehicles to analyze the reliability by part

inspection is one way to obtain real world usage data. Appropriate durability

testing in engine test cells or  eld test vehicles, preferably with accelerated

testing, may provide a good estimate of the failure rate to substitute service

reliability data. The current trend is to use numerical simulation to predict

failure rate in the early stage of the design. The simulation model needs to

be tuned by using durability test data or reliability service data.

A more detailed reliability model at a speci ed vehicle mileage or

operation time is the classical ‘stress–strength interference’ probabilistic

(nondeterministic) model (Fig. 1.13 in Chapter 1). The stress–strength

Diesel-Xin-02.indd 180 5/5/11 11:44:56 AM



181Durability and reliability in diesel engine system design

model was initially applied to fatigue reliability analysis by Freudenthal et

al. (1966). This model has been widely used to calculate the reliability of a

mechanical item when the associated stress and strength probability density

functions are known. The stress here is referred to the structural durability load

encountered in usage, and the strength is referred to the limit of the ability

of the part/material to resist failure. Unlike the time-based model in equation

2.14 where the random variable is time-to-failure, the stress–strength model

uses stress and strength as the random variables. The statistical variability

caused by noise factors such as product population, hardware tolerance,

usage style and environmental condition is represented. The stress–strength

model is usually in the following form:

Rf

SS

f

S

Rf

S

Rf

PD

Rf

PD

Rf

Fs

trengt

hP

SS

hP

SS

f

hP

f

DF

hPDFhP

stre

s

Rf =Rf

(

Rf (Rf

Rf

S

Rf (Rf

S

Rf

PD

(

PD

Rf

PD

Rf (Rf

PD

Rf

Fs

(

Fs

trengt

(

trengt

hP

(

hP

trengthPtrengt

(

trengthPtrengt

)d

SS)dSS

hP

)d

hP

SS

hP

SS)dSS

hP

SS

++

,,

f

,,

f

hP,,hP

f

hP

f

,,

f

hP

f

DF,,DF

hPDFhP,,hPDFhP

,,

Fs,,Fs

trengt,,trengt

hP,,hP

trengthPtrengt,,trengthPtrengt

(

,,

(

Fs

(

Fs,,Fs

(

Fs

trengt

(

trengt,,trengt

(

trengt

hP

(

hP,,hP

(

hP

trengthPtrengt

(

trengthPtrengt,,trengthPtrengt

(

trengthPtrengt

hP

)d

hP,,hP

)d

hP

••

++••++

++

ÚÚ

++

ÚÚ

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

ÚÚ

Rf

ÚÚ

Rf

S

ÚÚ

S

Rf

S

Rf

ÚÚ

Rf

S

Rf

Rf (Rf

ÚÚ

Rf (Rf

Rf

S

Rf (Rf

S

Rf

ÚÚ

Rf

S

Rf (Rf

S

Rf

–

ÚÚ

–

++

ÚÚ

++

•

ÚÚ

•

++••++

ÚÚ

++••++

È

++

È

++

ÚÚ

++

È

++

ÚÚ

++

ÚÚ

È

ÚÚ

Rf

ÚÚ

Rf

È

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

È

Rf (Rf

ÚÚ

Rf (Rf

++

ÚÚ

++

È

++

ÚÚ

++

++••++

ÚÚ

++••++

È

++••++

ÚÚ

++••++

Î

ÚÚ

Î

ÚÚ

Rf

ÚÚ

Rf

Î

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

Î

Rf (Rf

ÚÚ

Rf (Rf

ÚÚ

Rf (Rf

Í

Rf (Rf

ÚÚ

Rf (Rf

ÚÚ

Rf (Rf

È

Rf (Rf

ÚÚ

Rf (Rf

Í

Rf (Rf

ÚÚ

Rf (Rf

È

Rf (Rf

ÚÚ

Rf (Rf

Î

Í

Î

ÚÚ

Î

ÚÚ

Í

ÚÚ

Î

ÚÚ

Rf

ÚÚ

Rf

Î

Rf

ÚÚ

Rf

Í

Rf

ÚÚ

Rf

Î

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

Î

Rf (Rf

ÚÚ

Rf (Rf

Í

Rf (Rf

ÚÚ

Rf (Rf

Î

Rf (Rf

ÚÚ

Rf (Rf

˘

hP

˚

hP

˚

,,

˚

,,

hP,,hP

˚

hP,,hP

˙

hP

˙

hP

˘

˙

˘

hP

˚

hP

˙

hP

˚

hP

˚

˙

˚

,,

˚

,,

˙

,,

˚

,,

hP,,hP

˚

hP,,hP

˙

hP,,hP

˚

hP,,hP

sss

sP

ngt

ss

fx

PD

fx

PD

Fs

fx

Fs

tres

fx

tres

sP

fx

sP

fx

sP

fx

sP

DF

fx

DF

sPDFsP

fx

sPDFsP

stre

fx

stre

ngt

fx

ngt

h

fx

h

ngthngt

fx

ngthngt

x

()

ss()ss

d

ssdss

=

()

sP

()

sP

fx()fx

sP

fx

sP

()

sP

fx

sP

()

fx()fx

d

xdx

–

+

,,

sP,,sP

fx

,,

fx

Fs

fx

Fs,,Fs

fx

Fs

tres

fx

tres,,tres

fx

tres

sP

fx

sP,,sP

fx

sP

fx

,,

fx

sP

fx

sP,,sP

fx

sP

DF

fx

DF,,DF

fx

DF

sPDFsP

fx

sPDFsP,,sPDFsP

fx

sPDFsP

sP

()

sP,,sP

()

sP

fx

sP

()

sP

fx

sP,,sP

fx

sP

()

sP

fx

sP

•

Ú

–

Ú

–

2.15

where f

PDF,stress

(s) is the probability density function of the stress, and

f

PDF,strength

(S) is the probability density function of the strength. Note that

for the Beta distribution (see Table A.2(c) in the Appendix), the upper limit

of the integrals in equation 2.15 may be a  nite value (Kececioglu, 2003).

The reliability is de ned as the probability of stress less than strength. The

failure rate or risk of failure is de ned as the probability of strength less

than stress, essentially re ected by the overlapping area under the probability

density function curves of stress and strength. The reliability and the failure

rate are related by the following relationship:

R = 1 – P

failure

= 1 – P (S < s) 2.16

where P (S < s) indicates the probability of strength less than stress. The

reliability model can be applied to either a component or an entire system.

The degree of stress–strength interference is dependent on time because the

durability parameters such as accumulated damage and material strength are

related to time. A stress–strength model can be used to calculate reliability

values for different speci ed time as input. Then, the reliability data obtained

as such may form a time-based reliability model.

The stress–strength model contains the detailed design information.

Therefore, it should be the focus of reliability modeling in engine system

design. The probabilistic sensitivity analysis using such a stress–strength

model can identify the most important parameters that affect the structural

reliability.

The stress–strength interference model can be used at either steady-state

full load conditions or in transient driving cycles to evaluate the appropriate

design constraints that need to be used in system design such as peak cylinder

pressure or exhaust manifold gas temperature. In the transient cycle analysis,

Diesel-Xin-02.indd 181 5/5/11 11:44:57 AM



182 Diesel engine system design

after the time history of component temperature is obtained, the time history

of stress and strain can be calculated. When the failure can be evaluated based

on the load or stress (such as in the case of steady-state HCF problems),

stress may be used as the random variable in the stress–strength model.

However, in many other cases, the stress or a cycle average load may not

be the convenient or appropriate parameter to use. For example, in LCF the

dominating failure criterion is usually strain rather than stress. Moreover,

the load can come from either normal use or accidental misuse. The misuse

may only occur a dozen times at most during the lifetime of the engine so

that its load may not be representative enough. Furthermore, in transient

usage cycles a cycle-average stress or load also may not be representative

or conveniently available. In these cases, a more general damage–strength

interference model should be used as below:

Rf

SS

f

S

Rf

S

Rf

PD

Rf

PD

Rf

Fs

trengt

hD

SS

hD

SS

DP

f

DP

f

DF

DPDFDP

da

D

Rf =Rf

(

Rf (Rf

Rf

S

Rf (Rf

S

Rf

PD

(

PD

Rf

PD

Rf (Rf

PD

Rf

Fs

(

Fs

trengt

(

trengt

hD

(

hD

trengthDtrengt

(

trengthDtrengt

)d

SS)dSS

++

,,

f

,,

f

DP,,DP

f

DP

f

,,

f

DP

f

DF,,DF

DPDFDP,,DPDFDP

,,

Fs,,Fs

trengt,,trengt

hD,,hD

trengthDtrengt,,trengthDtrengt

DP,,DP

(

,,

(

Fs

(

Fs,,Fs

(

Fs

trengt

(

trengt,,trengt

(

trengt

hD

(

hD,,hD

(

hD

trengthDtrengt

(

trengthDtrengt,,trengthDtrengt

(

trengthDtrengt

)d

,,

)d

••

++••++

ÚÚ

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

ÚÚ

Rf

ÚÚ

Rf

S

ÚÚ

S

Rf

S

Rf

ÚÚ

Rf

S

Rf

Rf (Rf

ÚÚ

Rf (Rf

Rf

S

Rf (Rf

S

Rf

ÚÚ

Rf

S

Rf (Rf

S

Rf

–

ÚÚ

–

++

ÚÚ

++

•

ÚÚ

•

++••++

ÚÚ

++••++

È

++

È

++

ÚÚ

È

ÚÚ

Rf

ÚÚ

Rf

È

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

È

Rf (Rf

ÚÚ

Rf (Rf

++

ÚÚ

++

È

++

ÚÚ

++

++••++

ÚÚ

++••++

È

++••++

ÚÚ

++••++

Î

ÚÚ

Î

ÚÚ

Rf

ÚÚ

Rf

Î

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

Î

Rf (Rf

ÚÚ

Rf (Rf

ÚÚ

Rf (Rf

Í

Rf (Rf

ÚÚ

Rf (Rf

ÚÚ

Rf (Rf

È

Rf (Rf

ÚÚ

Rf (Rf

Í

Rf (Rf

ÚÚ

Rf (Rf

È

Rf (Rf

ÚÚ

Rf (Rf

Î

Í

Î

ÚÚ

Î

ÚÚ

Í

ÚÚ

Î

ÚÚ

Rf

ÚÚ

Rf

Î

Rf

ÚÚ

Rf

Í

Rf

ÚÚ

Rf

Î

Rf

ÚÚ

Rf

Rf (Rf

ÚÚ

Rf (Rf

Î

Rf (Rf

ÚÚ

Rf (Rf

Í

Rf (Rf

ÚÚ

Rf (Rf

Î

Rf (Rf

ÚÚ

Rf (Rf

˘

DP

˚

DP

˚

,,

˚

,,

DP,,DP

˚

DP,,DP

˙

DP

˙

DP

˘

˙

˘

DP

˚

DP

˙

DP

˚

DP

˚

˙

˚

,,

˚

,,

˙

,,

˚

,,

DP,,DP

˚

DP,,DP

˙

DP,,DP

˚

DP,,DP

mmm

age

DD

()

DD()DD

d

DDdDD

2.17

where D is a damage indicator, and S

D

is the strength in terms of critical

damage threshold for failure. The concept of damage was introduced in

Section 2.8.3, and the damage can refer to any failure. The value of S

D

is

usually around 1 for a given failure mode. The random variable of damage

may represent the cumulative damage due to all failure modes. The failure

rate due to all failure modes of the component or the system at a given target

mileage or operating hours is equal to the probability that the cumulated

damage exceeds the critical damage threshold. The strength distribution for the

critical damage threshold can be obtained by durability testing or numerical

simulation. Reliability target can be de ned for each component and each

failure mode in order to analyze them separately. Reliability target can also

be de ned for the entire engine system and a combination of all the failure

modes in order to characterize the overall system behavior.

The reliability analysis can be conducted by using a sampling method such

as the Monte Carlo simulation (to be detailed in Chapter 3). The probability

of failure can then be estimated by the ratio of the number of failures to the

total number of simulation samples. The probability density functions of

the stress, damage, and strength can be expressed by the normal, lognormal,

Weibull, or beta distributions. The method of using the stress–strength model

for reliability prediction based on stress pro les from real world usage was

discussed by Mettas (2005).

The probabilistic distribution of strength can be obtained experimentally,

for example by observing the failure mode. It may also be estimated with

probabilistic numerical simulation. In the case of partial failures in the test

(e.g., the crack propagation in the inter-valve bridge of the cylinder head

at a initial 5 mm crack length instead of the 10 mm full failure limit), the

strength needs to be derived by extrapolating the test result of partial failures

Diesel-Xin-02.indd 182 5/5/11 11:44:57 AM

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183Durability and reliability in diesel engine system design

to the time for which the component would have failed completely. An

example technique of such an extrapolation was demonstrated by Prince et

al. (2005) and Daubercies et al. (2009) by using their statistical model of

crack propagation rate.

The analysis of probabilistic distribution of stress or load in engine system

design needs to consider the worst case of damage to be encountered in engine

durability validation test and customer real world usage. The analysis needs

to consider both steady-state full load (e.g., several hundred hours running

at rated power) and transient cycles (e.g., thermal cycles). In system design,

most durability constraints are specied at full load, such as peak cylinder

pressure, exhaust manifold gas temperature and turbocharger speed. The

full-load operation affects the maximum pressure and temperature and their

occurrence frequency within a transient cycle. However, continuous full-load

operation does not necessarily mean more damage than that produced by a

transient driving cycle. Depending on the failure mechanism (e.g., LCF or

HCF), the contribution of full-load operation to component failure may be

different. For example, cylinder head durability is largely determined by the

inter-valve bridge temperature history due to LCF mechanism. In this case,

the thermal load cycling from low idle to rated power is more contributing

to component failure than continuous full load. Two other considerations

in selecting the pressure or temperature load prole in reliability analysis

are the effects of ambient condition change and the regeneration of diesel

particulate lter. For instance, the cylinder gas and exhaust temperature can

become hotter at part load when the engine air ow is throttled or post fuel

injection is applied to assist DPF regeneration. Morin et al. (2005) provided

an example of the adverse impact of DPF regeneration on the time history

of the cylinder head temperature and the associated thermo-mechanical

damage. The mechanical and thermal loads can be calculated using cycle

simulation, multi-body dynamics, and sometimes FEA. When transient metal

temperature history is required in the reliability analysis, thermal inertia of

the component should also be considered. The temperature time history needs

to be converted to mechanical stress history of the component in order to

compute the damage.

The integral in the stress–strength interference models can be calculated

by numerical integration (e.g., the Simpson’s rule) or the Monte Carlo

simulation (Kececioglu, 2003). If the probability density functions are in

normal distribution, the reliability may also be calculated analytically, as

elaborated by Kececioglu (2003) and Zhou and Yu (2006).

The reliability value is sensitive to the errors in the probability density

function curves and the relative scattering between the stress and strength

curves. Moreover, the reliability value is dependent on the selected condence

level and the number of tested samples. The estimate of the condence

interval for the predicted reliability or failure rate is important when the

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184 Diesel engine system design

strength distribution is obtained from a small sample of durability test data.

The con dence interval can be established by a Monte Carlo simulation of

the failure rate. A detailed procedure of determining the con dence level

was provided by Kececioglu (2003) and Daubercies et al. (2009).

The above interference models are one-dimensional models. Zhou and

Yu (2006) developed a two-dimensional stress–strength model by adding

another dimension to cover the variability of the stress asymmetric ratio.

They applied their model to diesel engine crankshaft reliability analysis.

The stress asymmetric ratio was de ned as the ratio of the minimum stress

to the maximum stress during a loading cycle. They illustrated that the two-

dimensional model could calculate reliability values more accurately than

the one-dimensional model.

Safety factors and safety margins

The design strength is usually established a priori based on past experience.

It must be high enough to cover the uncertainties and variability in operating

stress or load, material, and manufacturing processes. In the traditional

deterministic (i.e., single-valued) design approach, a safety margin is reserved

in order to achieve a reliable design. By de nition, the safety factor and safety

margin are arbitrary multipliers used to ensure the reliability of mechanical

items during the design phase (Dhillon, 2005).

The mean (central) safety factor is usually de ned as the ratio of mean

strength to mean stress. The mean safety factor is a good measure of safety

if the probability data spread of the stress and strength is not large (e.g.,

both having the normal distribution). The safety margin is de ned as the

safety factor minus one. Another commonly used safety factor is the ultimate

safety factor, de ned as the ratio of the ultimate strength of the material to

the working stress. Moreover, the extreme safety factor is the ratio of the

minimum strength to the maximum stress. In the deterministic approach, the

maximum stress is obtained by multiplying a factor by the nominal stress,

and the minimum strength is obtained by multiplying another factor by the

nominal strength. Then the extreme safety factor can be expressed in terms

of probabilistic distribution parameters as follows (Kececioglu, 2003):

f

sf

f

sf

f

,

=

– C

extreme

Sh

ss

S

sC

+sC +

s

2.18

where

S

and

s

are the mean values of strength and stress distributions,

respectively; s

Sh

and s

ss

are the standard deviations of the strength and stress

distributions, respectively; and C is a value selected by the designer (usually

between three and six).

Kececioglu (2003) illustrated that the concept of safety factor can be

fallacious because it may not properly re ect the reliability. For instance,

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185Durability and reliability in diesel engine system design

when the means of the stress and strength distributions are kept constant but

the standard deviations are varied, the overlapping area (i.e., failure rate)

of the two distribution curves will change, and this will indicate different

reliabilities. However, because the means of the distributions are unchanged,

the mean (central) safety factor remains the same and does not reect the

reliability change properly. Moreover, when both means and standard

deviations are changed, reliability obviously changes; however, the safety

factors may not respond to the change accordingly. This indicates the serious

deciency of the deterministic design approach by using safety factors in

reliability design.

A high value of the safety factor does not always ensure the system is

safe or reliable. On the other hand, it may cause over-design and excessive

cost. As illustrated by Kececioglu (2003) in his keynote summary on the

probabilistic design approach of design for reliability in robust engineering,

traditional deterministic design is not sufcient. A probabilistic design method

needs to be used. A more advanced and economical design approach based

on probability or reliability is more realistic for engine performance and

durability predictions. The nondeterministic probabilistic design methodology

enables the designer to design the component or system to a desired reliability

goal at a desired condence level.

2.13.5 Probabilistic reliability analysis in engine design

The tasks of reliability-based design optimization in diesel engine system

design generally include the following:

∑ correlate engine performance and durability test results with in-service

reliability data in order to develop accelerated testing and predict reliability

more effectively;

∑ conduct probabilistic simulation to check design sensitivity on

reliability;

∑ translate a reliability design target to performance and durability attributes

by using probability analysis;

∑ generate durability constraints for engine system design;

∑ optimize engine system reliability.

If the reliability does not meet the requirement, an optimization on material,

design, and load needs to be conducted to either increase the strength or

reduce the stress or damage. If the component engineer has to request a

change in the system loading parameters (e.g., pressure, temperature, heat

rejection), this request is passed to the system engineer to revise the system

design specications. Sometimes the stress can be reduced by the design

of another component. For example, a modication in engine coolant

passage may increase the heat rejection and reduce the cylinder head metal

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Xin Q. Diesel Engine System Design

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