Abstract—A tractable model for in-cylinder fluid motion

during the intake stroke is developed with particular attention

given to fluid flow through the intake ports. Due to innovations

in valve timing strategies for the 4-stroke internal combustion

engine, the fluid flow effects of different valve timings must be

quantified. An asynchronous valve timing strategy for

Homogeneous Charge Compression Ignition engines serves as

the motivation for this model development. For the purposes of

real-time engine control, the model is not a multi-zone CFD

model. Rather, the model divides the cylinder into two zones—

a mixed zone and an unmixed zone. The flow is modeled as an

intake jet of fluid that determines the rate at which hot exhaust

gas is transferred from the unmixed to the mixed zone. The size

of the mixed zone at the end of the intake process determines

how well the cylinder contents are mixed. The effects of the

valve timings on the flow velocity through each intake valve,

cylinder pressure, and temperature are also presented.

I. INTRODUCTION

ith concerns over environmental impact and upcoming

governmental regulation, the need for better

stewardship of our energy resources is becoming abundantly

clear. In the sector of personal transportation, where the

four-stroke internal combustion engine (ICE) is dominant,

there are two approaches to reducing environmental

impact—replacing the ICE or improving it. One technology

that fits into this second category is Homogeneous Charge

Compression Ignition (HCCI) [8].

HCCI is an advanced engine technology that promises a

15-20% improvement in the efficiency of existing, gasoline-

fueled, spark ignition (SI) engines [8,14]. Compared to a

diesel-fueled, compression ignition (CI) engine, nitrous

oxide emissions (NOx) are significantly reduced. HCCI

operates similarly to both SI and CI engines. During the

intake stroke, air and fuel are mixed in the combustion

chamber. As the air-fuel mixture is compressed during the

compression stroke, the temperature and pressure of the

mixture rises until it reaches the autoignition point for that

particular in-cylinder composition. The autoignition point is

a specific thermodynamic state for a combustible mixture

where, when this state is reached, the mixture will quickly

and completely combust. Since there is no flame front (as in

M. J. McCuen is with the Mechanical Engineering Department,

University of Minnesota, Minneapolis, MN 55455 USA.

Z. Sun (corresponding author) is with the Mechanical Engineering

Department, University of Minnesota, Minneapolis, MN 55455 USA.

(phone: 612-625-2107; fax: 612-626-1854; e-mail: zsun@umn.edu).

G. Zhu is with the Department of Mechanical Engineering, Michigan

State University, East Lansing, MI 48824 USA.

SI engines), combustion temperatures are lower, thus

reducing NOx emissions. The homogeneous nature of the

mixture lowers the particulate emissions.

HCCI is not without challenges. Arguably, the biggest

hurdle to overcome with HCCI is controlling combustion.

With HCCI, there is no distinct initiator (such as a spark or

fuel injection) for combustion. Combustion timing is

determined by the thermodynamic state at intake valve

closing (IVC). Therefore, to control combustion,

temperature, pressure, and composition must be controlled to

set the conditions at IVC.

One control method is residual affected HCCI. Hot,

residual exhaust gas is reused during the next engine cycle to

control the temperature of the mixture. The ratio of the hot

residual to the cool, fresh charge determines the overall

mean temperature at IVC [3,4,7,10]. There are a number of

strategies that can be used to enable residual affected HCCI,

including exhaust gas reinduction, exhaust gas recirculation,

and negative valve overlap—the strategy used in this

research. Negative valve overlap (NVO) traps exhaust gas

during the exhaust stroke of the engine by closing the

exhaust valves before the piston reaches top dead center

(TDC), thus preventing all of the exhaust from being

expelled from the cylinder.

Residual affected HCCI has its limitations—namely, a

constrained operating range. At high speeds and loads, there

is not enough time for residual and fresh charge to mix,

thereby creating fuel-rich regions that are subject to knock.

At low speeds and loads, large quantities of residual are

necessary to increase the temperature of the mixture to

appropriate values. With large amounts of residual, though,

mixing is poor, and the engine is likely to misfire. A

potential solution to these problems is to improve the charge-

residual mixing.

A strategy presented in [1,2] entails opening the intake

valves at different timings to create improved flow

characteristics, and therefore better mixing. The advantage

of this strategy is the ability to use NVO while avoiding the

inherent mixing problems associated with it. Thus, the

problems of NVO strategies (poor mixing) can be avoided

while still taking advantage of the benefits of NVO (such as

a reduced chance of piston-valve collisions in interference

engines). Furthermore, since the exhaust gas does not leave

the cylinder (as it does with reinduction strategies), there is

less heat lost to the environment and the overall efficiency of

the engine is greater.

Control-Oriented Mixing Model for Homogeneous Charge

Compression Ignition Engines

Matthew J. McCuen, Zongxuan Sun, and Guoming Zhu

W

2010 American Control ConferenceMarriott Waterfront, Baltimore, MD, USAJune 30-July 02, 2010

ThB19.5

978-1-4244-7427-1/10/$26.00 ©2010 AACC 3809

A unique advantage of this strategy is its ability to

improve mixing while using NVO. However, to fully utilize

this method, we must have accurate information about the

system behavior—particularly the fluid flow during the

mixing process. There are presently many publications that

use computational fluid dynamics (CFD) and detailed

numerical solvers to predict in-cylinder flow patterns. The

veracity of these simulations are not questioned by this work,

rather, they present a need to develop a more tractable model

for mixing that can be used for control system design. Since

the ultimate goal of this research is to implement a valve

control strategy for HCCI operation, a control-oriented

model of the system for real-time decision making and

control is required.

Fig. 1: Block diagram describing the overall behavior of the engine model

and control system.

By creating an accurate mathematical model for an engine,

the ability to design a fully flexible control system for the

engine is enhanced. While the focus of this research is

HCCI, the mathematical models can be used for other

combustion modes so that a truly flexible automotive

powertrain can be developed [11-13, 16]. Fully flexible

valve actuation, hybrid vehicles, and advanced transmissions

are enabling technologies for the overall goal of reduced

energy usage. Cylinder mixing is one component that, when

accurately modeled, allows the enabling technologies to

improve the performance of the overall system. For

example, through the development of this mixing model, we

will gain a better understanding of the influence of valve

timings on engine behavior. Armed with this information,

we can use a fully flexible valve actuation system to

implement the desired valve timings.

Fig. 1 describes the role of this fluid mixing model in the

overall development of a complete and accurate engine

model and control system. In Fig. 1, there is a desired

system behavior that is given to the Control System shown at

point (A). The output of the Control System, point (B),

gives various inputs to the engine. These parameters can be

the valve timings, the fuel injection timing and quantity,

spark timing, and the amount of exhaust gas recirculation

(EGR). The fluid mixing model calculates the degree of

mixing, and gives various parameters (mixture quality,

temperature, and composition) to the combustion model at

point (C). Forming a feedback structure, the output of the

combustion model (D), is sent back to the fluid mixing

model for use during the next cycle. Finally, the system

output is fed back to the control system at point (A). A

tractable and complete modeling method is necessary for the

development of the strategy shown in Fig. 1 for real-time

decision making and control.

II. NOVEL VALVE STRATEGY

As mentioned previously in section I, a novel valve

strategy proposed by [1] and [2] serves as motivation for the

development of this mixing model. A schematic of the novel

valve strategy is presented in fig. 2.

Fig. 2: Schematic of asynchronous valve strategy. The valve lift profiles on

the right depict two intake valves—one of which operates on a delay

compared to the other.

Typical valve strategies operate where all of the intake

valves have the same timing. This allows for the least

restriction of the flow and helps to reduce pumping losses.

However, as shown in [1], if the intake valves operate at

different timings, the motion in the cylinder is different.

More specifically, the amount of turbulence in the cylinder is

increased. One reason for this is that when there is only one

valve open, the velocity through that valve is increased.

Also, as the valves are opened and closed, pressure

differentials cause increased flow turbulence.

The analysis of the in-cylinder flow was performed with

CFD software in [1]. In later work [2], the authors used the

novel valve strategy in an engine simulation to determine the

effects of the timings on engine behavior. The authors

determined that because of the improved mixing afforded by

this valve strategy, the amount of residual exhaust gas that

can be used effectively is increased. Fig. 3, from [2], shows

the relationship between valve timings and amount of

residual.

3810

Fig. 3: From [2], this plot shows simulation results for the novel valve

strategy. For a given exhaust valve timing (y-axis), one intake valve

opening timing (x-axis) is kept symmetric, and the other intake valve is

swept from 0º to 110º. The contours show the amount of residual that can

be retained, with the novel valve strategy in black.

These results are promising, and demonstrate the need for

a mixing model. Real-time control development for this

strategy depends on a mathematical model for the underlying

processes within the cylinder. A tractable, yet complete

mixing model can describe the behavior of the fluid in the

cylinder for this strategy, and allow the control system to

determine the best set of valve timings to ensure that the

greatest amount of mixing is occurring.

III. MIXING MODEL

The goal of this model is to capture complex flow

characteristics in a complete and tractable model. Whereas

most models for mixing have a large number of

computational zones, a single-zone model, or, in this case, a

two-zone model, is desirable because of its relatively low

computational needs.

Modeling the flow processes with one or two zones can be

risky. Certain fluid mixing behaviors are best described by

CFD techniques. By reducing the dimensions of the model,

we risk ignoring important behaviors. However, knowledge

of overall flow behavior and its effects on combustion can

inform our development of a control-oriented model.

A two-zone model developed by [15] utilizes two distinct

regions for the intake process, and determines the chemical

composition of each region. However, the regions do not

interact with one another.

Another component of mixing occurs during the

compression stroke of the engine. The model presented in

this work does not account for that mixing because this

model was developed to explore the effects of the intake

flow (and valve events, in particular) on the overall degree of

mixing. Mixing during compression can be a significant

factor, but it lies beyond the scope of this model.

Fig. 4: Schematic of model design depicting two regions—one containing

a mixture of fresh charge and residual exhaust gas from the previous cycle,

and the other containing only residual. Inlet flow conditions cause the

regions to grow and shrink.

The model presented here, motivated by [5], is depicted

above in Fig. 4. The cylinder volume is divided into two

regions: a mixed region (Region 1) and a residual region

(Region 2). Until intake valve opening (IVO), the residual

region makes up the entire cylinder volume, and the volume

of the mixed region is zero. After IVO, the mixed region

grows (and the residual region shrinks correspondingly) at a

rate determined by the flow characteristics at the intake

valves. It must also be stressed that this model assumes two

intake valves per cylinder.

Region 1 is modeled as

rtmi mmm &&& +=1 (1)

where im& indicates the inlet flow, and rtmm& indicates the

rate at which mass in region 2 is transferred to region 1.

Similarly, region 2 is modeled as

rtmmm && −=2 . (2)

The size of region 1 is the quantity of interest, as it will be

used as an input for a combustion model.

The rate of growth of region 1, rtmm& , is a function of the

flow characteristics at the intake valves. Properties that alter

rtmm& , and therefore the mixing within the cylinder, are

velocity of the inlet flow and the quantity of residual mass

present in the cylinder. There are other factors that affect

cylinder mixing such as the angle of the inlet flow and

boundary layer behavior at the cylinder walls; however, this

work develops the model for inlet velocity behavior and

residual mass quantity.

The methodology for this method is as follows:

1. Determination of the flow entering the cylinder

2. Determination of the amount of residual in the

cylinder, and the effect on the system state when

the residual mixes with the fresh charge.

3811

3. Determination of the effect of inlet conditions and

system state on the growth of Region 1.

4. Application of mixing model to combustion

A. Flow through Intake Ports

The first stage in developing a mixing model is to model

the flow entering the cylinder through the intake ports. The

flow through the intake ports depends on the lift of the

valves, the geometry of the valves and ports, and pressure

changes caused by the valve orifice area and the moving

piston. A measure of the flow velocity that considers valve

and port geometry, as well as lift is known as pseudo flow

velocity [6]. This quantity, psv , is used instead of other

measures of flow velocity because of its unique ability to

capture the effects of port geometry as the inlet area changes.

θ

π

d

ds

A

Bv

m

ps4

2

= (3)

where B is the cylinder bore, Am is the minimum valve area,

and s is the distance from the piston pin to the crankshaft

center for a given crank angle θ . The relation for s is given

below:

s = a ⋅ cosθ + l2 − a

2 sin2 θ (4)

where a is the crank radius and l is the connecting rod

length. Am is a nonlinear function of the valve lift, Lv [6],

and takes into account the actual flow area due to flow

separation through the intake port. In (3), the pressure in

the cylinder does not appear explicitly. However, the

presence of θd

ds accounts for the piston movement and the

associated change in pressure.

Fig. 5: Lift profiles of intake valves used for this study. The maximum lift

is 5 mm. Valve 1 is open for 120 CAD and valve 2 is open for 125 CAD

The lift profiles used for the intake valves are simplified

profiles, an example of which is depicted in Fig. 5. The

valve strategy used sets the first intake valve to open at

θivo,1, and the second valve opens at a later timing, θivo,2 .

The inlet mass flow rate is also important to this model. It

determines the total amount of fresh charge that enters the

cylinder during the intake process. The inlet mass flow rate,

mi , is developed from the standard orifice equation [6]:

( ) ( )PPRT

PACm m

i

mmDi ,Ψ=

θ& (5)

Ψ Pm,P( )=

P

Pm

1

γ 2γ

γ −11−

P

Pm

γ −1

γ

12

,P

Pm

> 0.528

γ12

2

γ +1

γ +1

2 γ −1( ),

P

Pm

≤ 0.528

where Ti is the temperature of the fresh charge, Pm is the

pressure in the intake manifold, and γ is the specific heat

ratio. P is the pressure within the cylinder, and is given by:

( ) ciTmV

RP &&

θ

γ= (6)

where R is the gas constant, V θ( )is the cylinder volume at

any crank angle θ , and Tc is the average temperature in the

cylinder.

B. Residual and Fresh Charge Mixing

At the start of the intake process, the cylinder is entirely

residual gas. We use the ideal gas law to determine the mass

of residual gas in the cylinder at IVO,

( ) ( )rgf

ivoivor

RT

VPm

1,1, θθ= . (7)

P and V are the pressure and volume of the cylinder,

respectively, and are evaluated at IVO. Trgf is the

temperature of the residual exhaust gas and R is the gas

constant for the residual.

The average temperature in the cylinder during the intake

process, Tc , depends on the relative masses of fresh charge

and residual. While there are two-zones with distinct

temperatures during the intake process, the temperature Tc is

the average temperature over the entire cylinder. This is

because the pressure in the cylinder is uniform, and the

overall average must be used to compute the temperature. At

any point during the intake process, Tc is given by:

rvrivi

rgfrvriivi

ccmcm

TcmTcmT

,,

,,

+

+= (8)

3812

cv,r and cv,i are the specific heats for the residual and fresh

charge, respectively, and Ti is the temperature of the fresh

charge.

C. Determination of Region 1 Growth Rate

Determination of the rate of growth of Region 1 is the next

stage in this mixing model. Modeling the combustion

chamber as a piston-cylinder device (Fig. 6), where fictitious

divider divides Region 1 from Region 2, we can develop a

dynamic model of the system.

Fig. 6: Schematic of model for determining the rate of mass transfer from

Region 2 to Region 1. The intake flow into the cylinder is modeled as a

fluid jet that impinges on a mass-less piston. The force of the jet pushes

hot residual from Region 2 to Region 1.

In the model, the intake flow enters the cylinder and

behaves like a fluid jet. The velocity and area of the jet are

the total pseudo-velocity and total mean valve area,

respectively. The momentum of the fluid imparts a force on

the divider that divides the two regions.

Applying conservation of energy to the control volume

(Region 2), and solving for the mass flow out of Region 2,

the following is the result:

( )

prgf

IVO

rtmcT

VVPm

−−= 2

& . (9)

V2 is the volume of Region 2 at a crank angle θ as

determined by the ideal gas law, c p is the specific heat at

constant pressure, and VIVO is the volume of the cylinder at

IVO. It should be noted that V2 has two physical limits—it

is at its maximum size at IVO, where it is the entire cylinder

volume, and when all the gas is expelled from Region 2, the

region ceases to exist. P is the pressure exerted on Region 2

by the fluid jet.

P =ρv ps

2Am

Acyl

(10)

where Acyl is the cross-sectional area of the cylinder and ρ

is the fluid density. In (9), it is also assumed that the pressure

on both sides of the divider equalizes instantaneous, keeping

Region 1 and Region 2 in equilibrium. Thus, all flow energy

imparted by the intake flow is assumed to be partitioned into

work, and internal energy changes in Region 2 are ignored.

Combining equations (1), (9), and (10), the result for the

rate of change of Region 1 is

( )

prgfcyl

IVOmps

icTA

VVAvmm

−−=

2

2

1

ρ&& . (11)

D. Effect on Combustion

Once we are able to determine the size and mass of

Region 1, we can then see how it affects the combustion

process. The initial state at IVO is known. The mass of

Region 2 serves as a limiting factor. This occurs because

when all of the mass from Region 2 is expelled, the cylinder

is assumed to have perfect mixing, and Region 1 is the

cylinder volume. For the non-limited case, we are able to

use the ideal gas law to calculate the volume of Region 1 at

IVC. This volume, Veq , is then used in place of the actual

cylinder volume in the subsequent compression and

combustion phases. If any residual is left in Region 2, it will

still be compressed, but since the gas is inert, no reaction will

occur in Region 2. These phases use an Arrhenius integral to

compute the start of combustion. The Arrhenius equation [9]

is

( ) ( ) ϑϑθθ

θdRRAR

ivc∫=

2,

( ) ( )( )

−=

−

ivc

n

eqivcann

eqivc

n

ivcRT

EApRR

c

cϑν

ϑνϑ1

,

,2, exp

( )ϑ

νc

eq

eqivcV

V=, . (12)

A , Ea , n , and nc are system constants. ν is a volume

ratio used to simplify the equation. The inclusion of the

equivalent volume allows the effects of mixing to affect the

combustion timing and duration.

The temperature at IVC, Tivc , is taken from the

temperature of the gas mixture in Region 1. In a similar

fashion to (8), the temperature is

Tivc =micv,iTi + mrtmcv,rTrgf

micv,i + mrtmcv,r

(13)

where mrtm is the total mass transferred from Region 2 to

Region 1.

At the conclusion of the combustion process, the newly

combusted gas will mix with any residual gas that did not

transfer from Region 2 to Region 1 during the intake process.

Letting TH be the temperature of the newly combusted gas,

3813

the total mixture temperature is (allowing for instantaneous

mixing),

( )( )

rvrtmrvc

rgfrvrtmrHvc

excmmcm

TcmmTcmT

,1,,1

,1,,1

−+

−+= (14)

m1,c indicates the mass of Region 1 after combustion.

The air-fuel ratio, λ, of the mixed region is a function of

both the fresh charge and the excess air in the residual gas.

If the air-fuel ratio of the mixture at cycle i is defined as λ(i),

then the equivalence ratio at cycle i is defined as φ(i). φ*(i)

is the equivalence ratio of the fresh charge prior to mixing.

The air-fuel ratio of Region 1 after the intake process is

determined by

( )( )

( ) ( )( )

( )im

i

iim

iim

if

s

srtm

s

si

−+

−−+

+=

∗ 1

)1(1

)(ϕλ

ϕλ

ϕλ

λ

λ (15)

mi, mf, and mrtm are the mass of the intake charge, mass of the

injected fuel, and mass of the amount of residual transferred

from Region 2 to Region 1, respectively. sλ is the

stoichiometric air/fuel ratio.

IV. RESULTS

Simulating the intake flow velocity as a function of crank

angle (and therefore the valve timings) gives insight to the

behavior of the fluid. In Fig. 7, the pseudo-velocity through

each valve is shown with profiles similar to that of Fig. 5.

Fig. 7: Pseudo-velocity for the flow through each intake valve. The timings

for valve 1 are: open, 30° aTDC; close, 150° aTDC. For valve 2: open, 50°

aTDC; close, 175° aTDC. The maximum valve lift is 5 mm.

The peaks in Fig. 7 are the locations where the valves are

opening or closing. At these times, the open valve area

increases from zero (in the opening case) and decreases to

zero (in the closing case). These peaks are desirable as the

fluid jets are conducive to improved mixing.

Fig. 8: Pseudo-velocity for the flow through each intake valve. The timings

for valve 1 are: open, 30° aTDC; close, 150° aTDC. For valve 2: open, 80°

aTDC; close, 175° aTDC. The maximum valve lift is 5 mm.

Close to bottom dead center, the velocity approaches a

minimum. This is due to the engine piston moving slowly,

and therefore being unable to draw in as much fluid.

Similarly, the piston can draw the most fluid at 90º aTDC

(after Top Dead Center) as it is moving the fastest at that

point. This is illustrated by Fig. 8, where the second valve is

opened at 80º aTDC.

Fig. 9 shows the flow behavior of the intake flows where

each valve has a different maximum lift. Reduced lift

restricts the flow, which increases the velocity through the

port.

Fig. 9: Pseudo-velocity for the flow through each intake valve. The timings

for valve 1 are: open, 30° aTDC; close, 150° aTDC. The timings for valve

2 are: open, 50° aTDC; close, 175° aTDC. Valve 1 has a maximum lift of 5

mm and valve 2 has a maximum lift of 8 mm.

Fig. 10 shows the flow velocity into the cylinder if the

valves are treated as one variable area orifice.

3814

Fig. 10: Flow velocity for the total open intake area. The lift of each valve

is 5 mm. The period from 80 CAD to 150 CAD is where both valves are

open.

Comparing Fig. 10 to Figs. 7-9 shows that it is important

to study the effects of having multiple valves. Condensing

the information from multiple valves into a single area

obscures the effects that each valve has on the flow. In Fig.

10, where the second valve opens (80 CAD), the flow

velocity is reduced. This occurs because there is less

restriction (the open area has effectively doubled by opening

the second valve). However, the spike in flow velocity

through the second valve does not appear (as it does in fig. 8

at 80 CAD), even though the sharp increase in velocity is

important to overall cylinder fluid motion.

Figs. 11 and 12 show the cylinder pressure and

temperature, respectively, as a function of crank angle for the

two-zone mixing model.

Fig. 11: Cylinder pressure for asynchronous valve timing modeled by the

two-zone mixing model.

Fig. 12: Cylinder temperature for asynchronous valve timing modeled by

the two-zone mixing model.

Fig. 13 shows a simulation of the effect of valve timing on

the amount of mixing. Intake valve timings were swept from

20 degrees after TDC to 160 degrees after TDC. The

resulting volume of Region 2 (at the end of the intake

process) is presented in Fig. 13. With earlier IVO timings,

there is more time for mixing, and thus the final volume of

Region 2 is smaller than for later timings. Of particular

interest is the comparison between synchronous and

asynchronous valve strategies. For the asynchronous

strategy, the intake valve opening timings are separated by

10 degrees. This timing separation causes a noticeable

improvement in mixing, as shown by the decrease in the size

of Region 2 for a given valve timing. These results

correspond to data presented by [1,2] that an asynchronous

strategy can improve mixing. Furthermore, it is validation of

this mixing model as it can capture mixing effects produced

by different valve strategies.

Fig. 13: Comparison of synchronous and asynchronous intake valve

strategies. For the asynchronous strategy (red triangles), the valve opening

times are separated by 10 degrees.

3815

Fig. 14: Volume of Region 2 with an intake valve opening timing of 20

degrees aTDC. This represents a case where there is little residual gas in

the cylinder.

Fig. 15: Volume of Region 2 with an intake valve opening timing of 120

degrees aTDC. This represents a case where there is a large amount of

residual gas in the cylinder.

The behavior of the volume of Region 2 is shown in Figs.

14 and 15. In Fig. 14, the intake valves open 20 degrees

aTDC. This is equivalent to the case where there is little

residual gas in the cylinder. As shown by Fig. 14, by

approximately 40 degrees aTDC, all of the residual in

Region 2 is expelled. For Fig. 15, the intake valves are

opened late (120 degrees aTDC), signifying the case where

there is a large amount of residual in the cylinder. The

volume of Region 2 decreases, but at the end of the intake

process, there is still a quantity of residual remaining.

V. CONCLUSIONS AND FUTURE WORK

This paper presented a two-zone model for charge-

residual mixing in HCCI engines motivated by an

asynchronous valve strategy to extend the range of operation.

Simulation results indicate that the modeling method

proposed in Section III is valid method to pursue. While this

model cannot replicate the flow behavior with the accuracy

of CFD software, it is a less computationally demanding

proposal for quantifying the effects of valve timings on in-

cylinder mixing. We intend to verify this work by

comparing it to CFD simulations, as well as experimental

results. Then, the fluid mixing model will be incorporated

into a complete engine model as described in Fig. 1 of

Section I.

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