Pressurewaveturbocharging Final 1 [619786]

CHAPTER
Pressure Wave Turbocharging

1. Introduction.
One of the world’s major concerns is the rapid climate change due to the increasing levels of
greenhouse gas emissions, the main cause of global warming and its long -term dangerous
consequences [1]: melting of polar ice and rising seas levels, extreme weather, negative impact
on agriculture, forestry, energy and tourism, risks for wildlife, risks for human health,
heavy costs for the society and economy.
The European Union established as one of its k ey targets for 2020 a 20% reduction in greenhouse
gas emissions compared with 1990 , and a 40% reduction for 2030 [ 1]. Consequently, the EU has
taken legislative action to restrict higher levels of pollution, especially arising from burning the
fossil fuels .
The primary consumer of fossil fuels are the propulsi on systems, being stated [ 2] that the road
transportation sector is one of the main sources of greenhouse gas emissions, e.g. CO 2, and air
pollutants. Therefore, the internal combustion engines (ICE) have become the point of interest in
terms of reducing emissions, as well as efficient use of energy.
Among the measures to reduce vehicle emissions EU, the most common are: the introduction of
alternative propulsion technologies – by developing the hybrid or electric cars; eco -innovation;
improving the efficiency of conventional engines – by implementing advanced technologies, such
as: gasoline direct injection, variable valve lift and opening, deactivating cylinders, turbocharging;
implementing specific t echnologies for the exhaust system – by using oxidation catalysts, catalysts
substances injected into the flue gas stream, cellular filters, traps and absorbers, etc.
A well -conceived thermal management of the ICE can contribute in achieving the targets ou tlined
above, by:
● energy conservation and waste heat recovery,
● improving the overall efficiency and performance,
● lowering fuel consumption and, thus,
● reducing gas emissions.
Thereby, it follows a decreased dependence on fossil fuels, and therefore a co nservation of natural
resources and a reduction in the impact of propulsion systems on the biodiversity.
A highly prevalent manner to increase the power output and the engine’s efficiency is
supercharging, i.e. rising the intake manifold pressure by produc ing considerable boost for the
inlet fresh mixture or combustion air. This is achieved by using superchargers or turbochargers,
the equipments positioned within the intake manifold system.

Supercharging is about raising the engine’s charge density – air or fuel -air mixture – before filling
the cylinders. In this way, higher mass of charge can be compressed in each cylinder, containing
more oxygen available for combustion than conventional method of natural intake of fresh charge.
As a result, combustion w ill be more effective, thereby raising the generated power.
Increasing the pressure intake air is performed usually in a compressor that can be driven directly
from the crankshaft (supercharger) or by a tu rbine fixedly connected to the compressor shaft
(turbocharger), turbine driven in turn by the engine’s exhaust gases.
Supercharging aims to increase the torque and engine power by raising bmep while reducing the
maximum engine speed. Thus, the mechanical losses of the engine the noise and fuel
consumpt ion decreases, with the benefit of prolonging engine life [3].
Superchargers are efficient in low and medium engine speeds, but their improvement on the
thermal efficiency is very limited because the effective work is reduced with the amount consumed
to drive the compressor. On the other hand, turbochargers have higher thermal efficiency because
the compressor is driven by the energy coming from the exhaust gases, but they work properly on
the engine high -speeds range and they induce increased exhaust gas p ressures.

A particular type of “compressor” is the pressure wave device – known as “wave rotor” or
“pressure wave supercharger” (PWS) – that use also the energy of the combustion gases to induce
forced air into the admission manifold, but rely on the press ure waves action inside narrow
channels. The fast response on the engine performance for any of the engine speeds makes the
PWS a good option for supercharging the IC engines for road vehicles.

Pressure Wave Turbocharging.
Fundamentals of PWS. General Operating Principles. Design and Construction.

PW superchargers exploits the pressure waves to transfer energy from the combustion
gases to the air intake. The PWS basically consists of a rotor, also called “cell wheel” in which are
machined longitudinal narrow channels, positione d radially on one or two rows (Fig.1 a). Inside
the channels, PWS transfer energy directly from the exhaust gas towards the fresh air intake by
means of shockwaves, through short direct contact between fluids, without using any additional
mechanical elements ; the interaction between the high pressure hot gases and the low pressure
cold air induce boost .
In principle, the exhaust gases produce shockwaves that expands within the channels and
compress the intake fresh air. The process of equalizing pressures is extremely fast through the
channels; therefore the phenomenon of mixing between the two fluids is insignificant.

The cell wheel rotates inside a cylindrical steel casing, in between two fixed plates, placed at both
ends of the rotor ( Fig.2). The plates are provided with passages that allow the air and the gases to
flow towards or from the rotor channe ls. The air passages, connected to the intake manifold is
called the “cold stator” (Fig.1b) and the exhaust gas passages flow through the passages called
the “warm stator” (Fig.1c) . The cell wheel is driven by a separate electrical motor or by a belt
driven by the crankshaft.

(a) (b) (c)

Figure 1 – (a) rotor, also named cell wheel ; (b) warm stator ; (c) cold stator

Figure 2 – PWS construction elements [4]

The longitudinal channels shaped into the cell wheel are opened at the end, forming radial holes
arranged in angular staggered rows. It is considered that this configuration substantially reduces
– by interference – the whistling which was observed t o the original model with only a single row of
cells. The rotor is cantilever mounted, supported in its own shaft towards ball bearings; an
extension of the shaft after the bearings supports the pulley -belt drive mechanism that rotates the
cell wheel with a multiple of the engine speed (3 -4 times or other) [3].

Figure 3 – Four port PWS working fluids and assembly elements

The working inlet fluids for a PWS are: the high pressure and high temperature burned gases
discharged by the engine (HPG) and the low pressure and low temperature fresh air (LPA) to be
delivered into the admission manifold. After the acoustic and thermody namic processes action
through the channels, the outlet fluids leaving the PW supercharger are: the expanded low

pressure gases (LPG) evacuated by the exhaust system and the compressed high pressure air
(HPA) going towards the cylinders (Fig.3) . The inlet and outlet fluids circulate through well
designed “windows”, usually called ports, shaped in the rotor’s end plates. The designed
dimensions and form of ports, their number and position are conceived different for each
application.
The common c onfiguration with four ports consists on: two ports for the exhaust gases – one
port for HPG and one for LPG, the second designed much larger in order to extend the discharge
time of the exhaust gases towards the silencer system – as well as two ports for the air: one wide
port for LPA and a narrower one for HPA. These passages are connected to the inlet and outlet
manifolds, respectively.

PWS can be designed in two con figurations, as shown in
Fig. 4 :
a) passing -through flow (TF) – when the working flui ds
flow in the same direction and
b) reverse flow (RF) – when each fluid (gas or air)
enters and exits on the same side.

The through -flow rotor has both inlet ports (fresh air
and exhaust gases) placed on the same end side of rotor,
while the outlet port s (compressed air and expanded
exhaust gase s) are on the other side. The reverse -flow
configuration provide the air inlet and outlet ports on one
side, named the “air casing”, and the gas inlet and outlet
ports placed on the “gas casing”, allowing for both fluids
to flow in and out on the same side. Even though these
two configurations ensure similar performance
improvement, they are substantially different when
investigating their inner processes. In the four ports TF
rotor, both fluids – the cold air and the hot gases – travel
the rotor longitudinally, maintaining an average
temperature for the rotor body, relatively constant over
the entire length of the rotor. This allow a self cooling
effect to the rotor, making the TF configuration suitable
for gas tur bines applications, where gas temperatures are
elevated. The RF configuration, where the air casing
remain rather cold while the gases casing gets hot, was
used mostly in su percharging applications [ 5].

Figure 4 – Four port PWS configurations [ 4]
(a) The passing -through flow configuration
(b) The reverse flow configuration

The number and location of the ports, as well as the thermal energy source have made the
distinction between various applications of these devices, serving different purposes. For instance,
the four ports configuration was used for supercharging the internal combustion engines, three
ports design was applied to pressure separators or pressure equalizers, while rotors with two,
four, five or nine ports were integrated into gas turbines applic ations [ 5].

Essentially, as the cell wheel rotates, the channel openings are exposed alternatively to the inlet
or outlet passages, allowing the fluids flow through the ports. The compression and expansion
waves are thus initiated inside the rotor channels; the high pressure gases develop pressure
waves that evolve inside the channels and compress the intake charge.

Inside the cell wheel, the real energy exchange takes place at the speed of sound, that depends
on the exhaust gas temperature, that i s not dependent on the engine speed. Therefore, the
pressure wave effect is optimal for a single operating designed point. If the rotor is crankshaft belt
driven, there is a constant transmission ratio between ICE and PWS. Thus, the wave strength is
dimini shed when the operation conditions are on an off -design point. In order to overcome this
disadvantage, there have been designed “pockets” within the rotor housing that allow the PWS to
respond rapidly to the changes in engine operating requirements. The po ckets ensure good boost
pressure curves as well as high efficiency for a wide range of operating conditions [6].

In designing the wave rotor, many challenges had to be assumed and solved: number of
channels, shape, diameter, length, number and shape of cell rows, material of rotor and other
elements, leakage effects, noise, weight. For instance, the cells usually axial, straight shaped, can
be also curved, similar to the turbine blades. More, the number of channels affects the rotor
operation – as it increases, the entry flow losses increase. Also, because of the high centrifugal
forces, high frequencies shockwaves and cyclic temperature gradients, the rotor material has to
be chosen to handle all these stresses. More, by using ceramic materials, the r otor became
lighter, suitable for “free -running” (no belt driving necessary), reducing thus the additional load at
the crankshaft. In terms of noise, as the frequency lies in the audible zone for the rotational speed
range of a common wave rotor, the well -known “whistle” has to be eliminated. One method to
reduce the noise was to break the symmetry of the rotor cells or by using multiple f lutes (rows of
channels) [7].
Another important challenge in designing the wave rotor is the leakage that seriously affects the
performance. First, the rotor and the casing has to be built of materials with the same expansion
characteristics. To minimize the leakages, the clearance bet ween the rotor and the end plates has
to be minimized, yet preventing the occurrence of contact, regardless of the thermal regime.
Nevertheless, the ability to produce the wave rotor device with low costs, at large scale, have
solicited the designers t o find solutions for refining the manufacturing process.

2. Pressure Wave Turbochargers. History.

The concept of supercharging, respectively supplying an internal combustion engine with
pressurized air dates back to early last century. The oscillatory and thermodynamic phenomena
last since forever on Earth, but scientists have deciphered these processes in relatively recent
years and continue to study them. The turbulent movement of particles, the steady or non -steady
fluids flow, the p ulsatory phenomena action have shown their potential more since the beginning
of the 20th century. However, the laborious and detailed calculations necessary to study the non –
steady phenomenon inside fluids, have hindered progress to be made in this direct ion.
The beneficiaries of these complex and interconnected processes are the internal combustion
engines and other machines that can record improved performances whether these phenomena
are known and accurately described.
Basically, the concept that descri bes the operating principles of machines using the non -steady
flow is the transfer of energy by means of pressure waves. The wave pressure devices have some
common characteristic elements: a rotor with longitudinally shaped narrow channels within. The
ports machined into the end fixed plates control the working fluids flow from the manifold pipes
towards the pressure wave channels and vice -versa. The energy transfer is achieved through the
direct interaction between fluids by using the non -steady pressure w aves.
Unlike the steady -flow turbomachines that compress or expand the working fluids, the wave rotor
performs both compression and expansio n in the same component [5]. Despite the advantages of
the rotor wave machines comparing to turbomachines, such as the rapid response, the low speed,
the simple geometry translated into low fabrication costs, the reduced erosion of the channels and

the possibility of good self -cooling of the rotor, the pressure wave machines raised some
challenges that limited the large commercial implementation of these devices.
These difficulties were of mechanical nature – like sealing and thermal – and of theoretical nature –
understanding and defi ning the complex non -steady flow phenomenon was the main factor that
hampered the pressure wave machines development. During the last century, the efforts to
improve the geometry and functioning features of the rotor wave, accordingly to a certain
applicat ion, were sustained by the continuous progress made in solving the equations describing
the inner nonlinear phenomena and in implementing the new techniques and technologies. Also, a
continuous impetus for achieving energy efficiency, for withdrawing the o ldest technology and for
responding permanently to the market changes have stimulated new interest in wave rotor
technology .

Wave engines with internal combustion, rotary thermal separators, such as wave rotor
refrigerators, the shock tunnels, pulse deto nation engines, wave rotors and its best known device,
the Comprex, which was developed as a replacement for conventional turbochargers, etc., are
some applications using the pressure wave compression and unsteady -flow phenomena [5].
The first pressure exchanger prese ntation dates from 1906 [8] when Knauff patented a cell drum
called ‘semi -static pressure exchanger” (SSPE) because its pressure characteristics are almost
entirely independent of its speed [9]. This device was initially described as a rotor with curved
rotor blades and inclined stator nozzles that provided output shaft power, therefore showing a
pressure exchange engine, using a steady -flow energy transfer by mixing compressible or
incompressible fluids [9]. In 1913 a German engineer, Burgha rd patented a device resembling
with the actual cell wheel . His device was a rotating drum with axial channels shaped on the
periphery, as a continuous source of pressurized air [ 7]. As the knowledgement on the unsteady
flow field was insufficient, his device was never developed. In 1928 another SSPE was proposed
by Lebre [10] for a refrigerating unit, both with a geometry implying long narrow channels.
Around 1928, Burghard recognized the process of uns teady -flow transfer in compressible fluids
using the pressure wave process, and patented a simpler device [11 in 9 ] known as ‘dynamic’
pressure exchangers (DPE) to distinguish it from previous "static pressure exchangers”. Inside
DPE the pressure waves ar e used in both compression and expansion processes eventuating
within the rotor channels. This gave the designation of these devices as “wave rotors”.
Inspired by Burghard patent, researchers have sought for decades solutions for the development
of the wav e rotors concept, being limited by the difficulties mainly related to poor understanding
of unsteady -flow processes.

Some inventions related to the wave phenomenon were patented in the same period in different
countries. In 1933 Michael Kadenacy (UK) tuned the engine exhaust pipes using the pressure
wave effect, later called “Kadenacy effect”,, obtaining considerable power output, but in a narrow
range of speeds. Around 1938, Johann Wydler (USA) designed a rotating device that used the
exhaust g ases energy for supercharging a reciprocating engine. Prof. Arthur R. Kantrowitz at
Cornell University designed a device working with high pressure ratio compression and expansion
waves. His efforts were thwarted by the mechanical difficulties occurred. Ho wever, at Cornell
Aeronautical Laboratory CAL the non -steady flow concept was developed over the years, having
an important contribution in developing of energy exchangers for gas turbine and various
stationary power applications [12 in 5].
During the Worl d War II, in 1940, Claude Seippel of the Brown Boveri Co., trying to apply the
Lebre principle to a heat pump, acknowledged that the pressure waves can efficiently transmit
energy, i.e. the expansion of a gas towards another gas to be compressed [13], both gases being
in direct contact. The first machine using the wave rotor concept was implemented in Switzerland
by Seippel as a high -pressure stage for a gas -turbine locomotive engine.
Seippel coined the term “COMPREX” due to COMPression – EXpansion processes that take place
within the rotor channels.

The rotor had 30 channels shaped within, rotating with 6000rpm. Two ports on each side of the
rotor allowed the passing of gas and air. T he pressure ratio was initially 3:1 with a global
efficiency of 69%, as reported in the tests performed in 1941 -1943 [5]. Accord ing to Seippel’s
patents [14-17], a power boost of 80% and a 25% increase in performance were expected. Even
though the first rotor worked satisfactory, proving the potential of pressure waves in transferring
energy, when installed on an engine, it showed that improvement in designing and matching need
to be done. The device was replaced by a heat exchanger which yielded a somewhat higher overall
thermal efficiency. No further development work is known to have been made [9].

However, the work of these researchers outlined the idea of using the pressure wave rotor for
turbocharging diesel engines. First attempt in implementing this new concept and to gather
enough data to establish a correlation between measured performance and theory, was made by
ITE Circuit Breaker Co., Philadelphia, USA, under the supervision of Prof. Arthur R. Kantrowitz and
support of the Bureau of Aeronautics [13]. The first units were designed and bui lt starting with
1949. In 1951 came the first encouragi ng results. An overall pressure ratio of 4.5 and an overall
thermal efficiency of 16% were achieved on a small test rotor, 4 in in diameter and 6 in long [9].
In 1954 were run the first tests that proved that pressure wave superchargers can be used for
supercharging the diesel engines. In 1955 a cooperative program between ITE Circuit Breaker Co.
and Caterpillar Tractor Co. started, resulting a small PWS prototy pe that incorporated all the
experience gained. ITE continued to perform tests until 1957 when started the tests on vehicle
[13], using the so -called device COMPREX* as a diesel -engine supercharger. The early version of
the supercharger did not yield suffi cient manifold pressure at very low engine speed, at which the
clutch is engaged [9]. However, the tests proved that the wave rotor is a simple device, it can
deliver high air density over a wide speed range and it allows rapid load changes with no lag or
smoke [13].
A cooperative program between ITE and BBC Brown Boveri & Co. started in 1955 as a result of the
promising results obtained. BBC Brown Boveri & Co., as a turbochargers producer, continued to
develop pressure -wave superchargers for diesel engines [18 in 5], collaborating with the Swiss
Federal Institute of Technology ETH Zurich. They succeded to make cycle modifications to
overcome the deficiency of the early version of superchargers [9]. Meanwhile, two distinct
directions of research were approached: gas turbines and enhancement of ICE performance.
Market studies and research at BBC showed that the second field had a higher market value and
hence the effort towards engine supercharging had a greater support.
Around the same time, the British company Power Jets Ltd. started to work on multiple wave rotor
applications, initially on IC supercharging but later on gas turbines, pressure equalizers, air cycle
refrigeration devices. For instance, the hungarian engineer Jendrassik worked at Power Jets Ltd. to
develop the rotor wave for aircraft engines applications, proposing around 1949 the wave rotor as
a high pressure topping stage for ear ly aircraft engines [5].

Also, in the mid 50’s, a new rotor geometry was designed by Pearson at Ruston -Hornsby Turbine
Company in UK, producer of diesel engines and gas turbines. The device called the Pearson Rotor
used its helical channels to change the path of waves and to produ ce shaft work the same way a
conventional gas turbine would. The rotor was 23 cm diameter and 7.6 cm length, and, despite its
reduced dimensions, it realized a single cycle per each rotation [19 in 5]. The Pearson Rotor was
known as the wave turbine engine and worked successfully in a wide range of operating conditions
e.g. 3000 –18,000 rpm, producing up to 26 kW at its design point with a cycle peak temperature of
1070 K and a thermal efficiency of around 10% [5]. The Pearson Rotor encountered a series of
difficulties, such as sealing against leakage and incomplete scavenging. The design provided
additional ports and injection nozzles to control undesired reflected waves, additional sealing and
well matched bearings taking into account the rotor thermal stre ss. Unfortunately, the motor was
destroyed due to excessive speed and its further development ended abruptly, invoking financial
difficulties of the company. The Pearson rotor represented a notable design of a device made for
producing significant power ou tput, being in the same time a successful pressure exchanger.

In late 1955 at Cornell Aeronautical Laboratory, INc. , Buffalo, New York was initiated a study of a
high-stagnation -temperature testing unit, that put the premises of developing a device based on
shock -tubes principles, which could produce a continuous stream of high -temperature, high –
velocity air, device ca lled the CAL Wave Superheater [20]. The validated results of the small scale
Wave Superheater led to the development of the 9000°R Wave Superheater, including a
requirement for an operational Wave Superheater Hypersonic Tunnel. The working principle
involv es the use of a group of shock tubes mounted on the periphery of a rotating drum, to
generate high temperature, high density and high velocity air. It requires a controlled flow of pre –
heated helium driver gas to provide a steady stream of high -enthalpy fo r the hypersonic wind
tunnel, a controlled flow of pre -heated charge air (which has to be superheated prior to expansion
to more than 4000 K and up to 120 atm for run times as long as 15 s), a controlled flow of pre –
heated prime helium gas and a controlled flow of coolant helium gas [20]. The CAL Wave
Superheater remained as a remarkable proof of the capabilities of wave rotor devices.

Klapproth of General Electric Company (GE) and Goldstein of NASA, in the late 1950s, initiated
conceptual and feasibility studies of combu stion in rotating devices [22-24 in 21]. The studies
considered stationary power turbines (with and without regeneration), and turbojets and ramjets
operating at various flight speeds. The authors reported improved efficiency for the engines in
certain operating conditions and emphasized concerns about and ignition timing with lean air –fuel
mixtures and off -design speed conditions for different fuel in puts and pressure outputs [21].

Between 1956 and 1963 GE conducted investigations on both pressure -exchange wave rotors and
wave rotor combustors. Klapproth & co. of the GE Ohio branch focused mostly on designing and
testing two experimental four -port pressure -exchange wave engine with s traight and curved
channels [21], a wave engine using air -gap seals. Some difficulties, such as inaccurate flow
calculations and a lack of attention given to the inner reflected waves resulted in a lower power
output as was predicted [36 in 5]. However, the Klapproth rotor clearly proved the possibility of
complete energy transfer within a wave rotor.

Therewith, in California, GE focused on designing of a prototype wave engine with curved channels
and co mbustion inside the rotor [21]. This configuration eliminates the external combustion
chamber, resulting a lower weight and a compact size. Between 1956 -1958 the experience gained
was applied in fabrication of the first internal combustion wave rotor, tested at California Avansate
Propul sion Syst ems Operation CAPSO of GE [25]. As described by Weber [25], the only test
performed on the wave rotor combustor lasted approximately 20 s when thermal expansion of the
rotor exceeded the tight seal clearances between the rotor and end plates and t hus the rotor
seized between the end plates. The test pointed on the difficulty in controlling the clearance
between the end plates and rotor because of the element’s thermal expansion. Despite the leaks,
a respectable overall pressure ratio of the wave ro tor was achieved of 1.2 to 1.3, corresponding to
global temperature ratios of 1.9 -2.6 measured between inlet and outlet ports of low pressure [5].
GE continued further testing only on p ressure exchange wave rotors between 1960 -1961. In 1963
the company dis continued the development of the wave rotor, reportedly due to changes in
business strategy [26].

In the mid 60s’, an attempt to use the experience of predecessors Klaproth and Pearson was
conducted by General Power Corporation GPC by initiating a program for vehicle application using
wave rotors [19 in 5]. Supported initially by Ford Motor Company and later by the Department of
Energy DOE and the Defense Advanced Research Projects Agency DARPA, the GPC rotor was
slightly different than the prede cessors rotor configurations [18 in 5]: (excessively) curved blades
and, in contrast to Pearson engine, no impulsive loading of the rotor blades from inlet manifolds to
produce power. Although GPC developed a computer code to obviate manual wave pattern de sign,

accurate calculations were still tedious. In early 80’s, Ford Motor Company withdrew its support
[27 in 5] and GPC ended the development of its wave engine.

D. B. Spalding of Imperial College London continued the theoretical and ex perimental work of
Jendrassik from Power Jets Ltd., elaborating the first wave rotor calculation methods that took
into account the heat and friction losses and their effect. The numerical model involved novel
features to ensure solutions free from instabi lities and physical improbabilities [28]. A computer
program was developed by Jonsson [29 in 5 ] using Spalding’s theoretical work, and it was applied
successfully to different pressure exchangers further investigations.

An interest in researching the wav e rotor have shown, also in the mid 60s’, Rolls -Royce in the UK,
by cooperating with Brown Boveri & Co. and with Berchtold of ETH Zurich și Spalding of Imperial
College London as consultants [30 in 5]. Their efforts were focused on developing a design a wave
rotor as a topping stage for a small helicopter engine Allison Mo del 250 , which utilized a reverse –
flow configuration rotor incorporated into a single turbine engine. The difficulties in designing and
quickly performing changes, as well as other chall enges, like leakage, start -up, bearing durability,
fuel control, affected the device performance. The program was canceled in 1972 against the
background of financial difficulties [31].

In 1971 the first COMPREX prototype was installed on a truck engine by the company Valmet
Tractors from Finland, while around the same year Mercedes -Benz started testing the COMPREX
for their passenger diesel cars. In 1979, BBC tested a version that was never implemented of
COMPREX for Formula 1 racing cars.
Diesel engines ’ supercharging using pressure wave rotors became of big interest starting with
1978 when car manufacturers like Opel, Mazda, Mercedes -Benz, Peugeot, Ferrari focused their
attention on COMPREX supercharging. The first success that became a mass production was the
2.3 litre supercharged engine on the Opel Senator model.
The first extended application of Comprex® on vehicles was in 1987 on a 2.0 litre supercharged
engine on Mazda 626 Capella model [32]. Mazda sold more than 150,000 units and is considered
one of the most successful commercial applications of PW technology. By the end of 80’s, Asea
Brown Boveri (former BBC) transferred the wave rotor development to Mazda, in Japan, when
researchers at ABB returned to the idea of utilizing wave rotor technology for ga s turbine
applications [5].

At the end of 70s’, Mathematical Science Northwest Inc. with the support of DOE and DARPA,
investigated the application of wave machines as a high pressure, high temperature top stage for
a gas turbine topping -steam bottoming cycle using coal -derived fuels; as a high temperature air
compressor for a coal -burning power plant; and as a "dirty" gas expander/air compressor for
pressurized, fluidized bed fired coal -burning power plants. The work implied the developing of a
laboratory wave rotor, its configuration including 100 channels, a 45 cm diameter and 40 cm
length, four ports, two additional small ports provided for more uniform port flows a nd a higher
transfer efficiency . The projections of energy exchanger performance were made using a computer
program, FLOW, that was developed for modelling the one -dimension al unsteady flows. The code
included the wall friction and heat transfer processes using correlations for steady, turbulent pipe
flows [33].

The experimental results were very well correlated to the numerical computational ones, for a
large range of oper ating conditions. The FLOW code used the flux -corrected transport algorithm
solving Euler equations accounting for heat transfer, viscosity, gradual port opening, and flow
leakage [5]. The influence of speed, port configuration and channel geometry, leakag e and heat
transfer on wave rotor performance was analyzed for on -design, as well as off -design conditions.

The flow modeling was probably the most critical in developing an accurate fluid dynamic
representation of the dominant unsteady flow processes and the principal losses, many of which
are two – or three -dimensional in nature . Configuration changes during the tests included variation
of the clearance between rotor and end walls, configuration of the driven gas inlet and outlet
ports, and increasing the area of the main driven stream outlet port. Operating parameters that
were varied were: the driver and driven gas inlet pressures, driven gas outlet pressure, rotor
speed, and flow rates through the port [33].

The first test conclusion reported by Th ayer was that decreasing the clearance between the rotor
and stationary port faces reduced the leakage and substantially improved the efficiency, therefore
leakage was recognized as a key problem for efficient wave rotor operation. Also, the experiments
showed that efficiency was relatively constant at pressure ratios less than approximately 0.8 and
it dropped quite rapidly at higher pressure ratios, due to a decrease in mass flow rate through the
high pressure outlet port. Another conclusion validated by t ests was that a major increase in
efficiency of the device can be achieved if the width of the driven gas outlet manifold was
increased. Modifying the low pressu re port widths and locations or moving the high pressure
ports relative to one another was pro jected by using the FLOW computer code to have a similar
effect. The data obtained from tests on MSNW wave rotor and the computed analysis led to a
much thorough understanding of the operation of real energy exchangers and the wave
mechanisms [33].
The MS NW wave rotor testing was discontinued in the mid 1980s, for reasons not reported [5].

Efforts on numerical simulation were reported also during the conference hosted by the Naval
Postgraduate School, Monterey, California in March, 1985, the ONR/NAVAIR Wave Rotor Research
and Technology Workshop. According to Atul Mathur [34], an interest in developing and
understanding of the basic flow processes and potential applications of pressure exchangers or
wave engines prompted the initiation of a research effort at the Turbopropulsion Laboratory of the
Naval Postgraduate School. The research direct ions were: development of an unsteady flow code,
together with numerical studies and experimental work.

Mathur introduced a numerical one -dimensional code, using the Random Choice Method, based on
the solution of Riemann problems. The method solves the g overning Euler equations in one –
dimension by solving a sequence of adjacent Riemann problems, specified as the initial conditions
for each succeeding time step [34]. The code proved to be useful for more rapid computational
calculations of the unsteady flo w process inside the wave rotor and of some preliminary design
features. However, it was not recommended for describing the real flow processes that involve
friction and heat losses, or for multi -dimensional flows [35].
The results of the uni -dimensional code were used in another program, called ENGINE, for
turbofan jet engine perfo rmance calculations [5].

A bi-dimensional code, based on Godunov method, was developed by S.Eidelman showing that the
dynamics of the passage opening significantly affects the flow pattern within the wave rotor
devices. In short terms, the code showed that it is essential to take into account the gradual
opening of the passages when designing a wave machine, for a proper timing, number and
geometry of passages, and also because of the losses occurring due to mixin g and wave
reflections [ 5, 37].

Also, around 1985, at Cornell University, researches on wave rotor were resumed under
supervision of Resler, former member of CAL Wave Superheater team . The direction approached
was mainly on elaborating new analytical methods and con cepts of three -ports rotor wave , five-
ports rotors , double -wave rotors , etc. [5]. The idea of increasing the pressure ratio and thus the
efficiency by using a compound unit, consisting of two or more wave rotors, one being the

supercharger for the other, was sugge sted first by Muller in 1954 [ 38 in 9]. This arrangement
would require on ly one power turbine. The cycle diagram of such an arrangement, incorporating a
heat exchanger, is shown in Fig. … (vezi fig.15 in [9] sau fig.10 in [5])

As synthesized in [5], during the 90’s other researchers became interested in investigating and
developing numerical methods to describe processes inside pressure wave machines: Eldin et al.
of University Wuppertal, Germany (numerical method based on characteristics theory), Pie chna et
al. of Warsaw University of Technology (one – and two -dimensional numerical codes), Oguri et al.
at Sophia University in Japan, Guzzella et al. at ETH Switzerland (a control -oriented model for
pressure wave devices used for engine supercharging, wit h accent on transient exhaust gas
recirculation modeling), as well as a diesel NOx emissions investigation under Comprex®
supercharging performed in Turkey.

Since the 90’s, significant progress was made in computer and automation fields, therefore,
analyt ical as well as experimental researches got a tremendous help, becoming easier to perform,
to model, simulate and to process the data results. A significant number of studies revealed a real
potential of pressure wave devices together with the improvements that need to be done in order
to obtain higher efficiency or overall performance.

The National Aerautics and Spatial Administration, NASA, was interested in studying the wave
rotor combustors for possible performance improvements of aircraft propulsion s ystems. Daniel E.
Paxson at NASA Glenn Research Center developed a wave rotor simulation code, experimentally
validated, i.e. a quasi -one-dimensional, time -accurate, reactive, CFD Euler solver, to provide
realistic results as it capture the complex gas dyn amics of the cycle, to calculate the wave rotor
geometry and to allow straightforward examination of many cycle modifications that can affect the
design a nd off -design performance [ 39, 40 in 21]. The code is recognized as a general analysis
tool for wave rotors [21].

Numerical integration is done by using a second order accurate, Lax – Wendroff based scheme,
utilizing Roe’s approximate Riemann solver to obtain unsteady flow domain for time varia ble
operating conditions. The scheme has been tested extensively with detonative calculations [ 3].
Paxson continued to improve the 1 -D model, based on the experimental results.
In 1994, Nalim at NASA has resumed the on -rotor-combustion concept, studying the ignition
process, flame propagation rates in various fresh charge mixtures, and different possible wave
cycles [ 21], extending the existing one -dimensional modelling code with a combustion modelling
part, to study wave cycles involving both detonation and turbulent -deflagration modes of
combustion [41 in Akbari 2]. The code’s capability for detonation prediction was widely used later
for the study of pulse detonation engines [ 41, 42 in 21].

Comprex® developed by BBC in mass production implied also solving the shortcomings resulted,
such as leakage, noise and rotor unequal heating: the rotor was mounted within a pressurized
casing to limit the leakage and was produced by materials with low thermal expansion coefficient
[36 in 5]. The performance was also improved by covering a wide engine speed range through
pockets shaped within the end plates in order to control the reflected waves . The result was a
reliable product for its purpose: ICE supercharging.

Even though the Comprex® had shown a huge potential, it could not overpower the conventional
turbocharger. In the early 90’s high and efficient supercharging was not known. At that time,
Swissauto WENKO AG started the development of the Swa tch-Mobile, today named Smart, and in
1994, as they realized that engine downsizing and power output increasing by supercharging are
the future solution in automotive industry, they stepped into pressure wave charger development.
Implementing the experienc e and research results of the predecessors regarding the pressure
wave charging principles, Swissauto company produced a more complicated version of the

pressure wave compressor, named HYPREX® . Swissauto WENKO AG drew up all the patent
applications regardi ng the relation of pressure wave charging for spark -ignition engine use, in
spite of close collaboration with Comprex AG Company [swissauto site]. The difficulties occurred in
developing the pressure wave charger for gasoline engines were related to the ap propriate
adjustment of the charger to the wide operation map of these engines. Hyprex was further
developed to our days because of its thermodynamical advantages, reliability and its direct
responding speed, making it ideal for charging small displacement gasoline engines. The benefits
of Hyprex imply boost response time and increased pressure ratios [swissauto site], lower specific
fuel consumption and reduced noise and emissions [ 43].

Starting with 1989, Asea Brown Boveri company , started a research project regarding wave rotors
for power -generation gas turbine applications [ 21], using at first a pressure -exchanger with
external combustion. The promising results of the studies obtained encouraged ABB to investigate
in 1991 the possibility of developin g a wave rotor with integrated combustion. A rotary -valved
single -channel wave rotor was built and tested in collaboration with the Swiss Federal Institute of
Technology Zurich, followed in 1992 by the design of an on combustion rotating wave rotor with
36 channels [44, 45 in 5]. For both models spark plug and hot gas -injection self -sustaining
ignition methods were utilized [191], making thus the combustion a continuous process. During
the tests various fuels were used and the fuel mixtures were stratified by means of four injectio n
nozzles. The prototype engine tests revealed some shortcomings that needed remedies, such as
gases removal, air cooling of engine and mechanical controlling of thermal expansion. The project
ended in 1994.

Since the beginning of the 90s’, as reported b y [5], a number of universities oriented their
researches towards the wave rotor technology and its applications. University of Florida initiated
numerical and analytical methods to investigate certain wave rotor geometry variations and its
design. At Purd ue School of Engineering and Technology, IUPUI, Nalim concentrated on the on –
combustion wave rotor, coming with a valuable concepts in optimizing the thermal management
of the inner processes: deflagrative combustion with longitudi nal fuel stratification [ 46 in 21], as
well as radial stratification [ 47 in 5], others. After 2000, at University of Tokyo, Nagashima and
co. developed 1 -D and 2 -D numerical codes that can simulate the flow behavior inside the rotor
channels; also, the group investigated the rotor wave application in ultra -micro gas turbines [48,
49 in 5]. Since 2002, Michigan State University has been preoccupied by pressure wave
technology application, such as wave rotor microturbines, refrigeration cyc les using R718, ultra –
micro – gas-turbines, wave rotors with radial flux (in collaboration with Warsaw University of
Technology, Poland). MSU together with the polish university developed also the FLUENT software
[50-52] used for the investigation of the ga so-dynamic phenomena inside the rotors with axial
and radial fluxes.

3. PWS operating principles.

The wave technology, as shown in the section above, has sparked since 1906 the interest of
researchers and manufacturers. In more than a century, thanks to the development and
implementation of new technologies, new reliable high -strength and high -temperat ure resistant
materials and nevertheless of the computational performances, workable wave machineries could
be realized and improved. Comparing to other engineering and production fields, the wave
technology registered relatively slow progress, though, con siderable results were achieved
especially on steady -flow tur bomachineries, despite the well -recognized better efficiency of non –
steady flow machines using the shock compression, process more ef ficient than the isentropic one .

Over the years, one of the wa ve technology usage, the wave rotor concept called Comprex,
developed by ABB company (former Brown Bovery & Co) has found its important application in
ICE supercharging.
The underlying principle of wave machines is the exchange of thermal energy in favor o f pressure,
by direct contact between the fluids and with no means of mechanical components.
The pressure wave supercharger is actually a pressure exchanger – the device using the shock
waves propagating within a time -dependant gas flow inside a tube to ch ange the pressure values.
The time variation is provided by the rotation of the device, that allows the tubes openings to pass
sequentially in front of the inlet and outlet ports [ 25].
The pressure exchanger is a simple device that exceeds the conventional super – or turbochargers
shortcomings, having the great potential of changing the pressure values in a very short time and
therefore, of tolerating transitory pick pressure and temperature values.
As described in the first section of this Chapter, the PWS transmits the energy for the engine’s
supercharging from the exhaust gases to the fresh intake air. The energy exchange takes place at
the speed of sound, by means of waves propagating inside the narrow PWS’ channels. As the PWS
is rotating, the openings s haped within the fixed end plates allow the successive alignment of the
ends of the channels in front of the fluid's inlet and outlet ports. The PWS geometry, dimensions,
ports’ opening timing and rotational speed need to be correlated for better results i n engine power
output and overall performance.

~ de inserat desenul clar, cu specificarea (a)… (d) in dreptul fiecarei faze ~
Fig. The PWS operating phases (a-d)

According to [ 3], the operating cycle of a wave supercharger can be conventionally divided into
four phases:
(i) fresh charge compression inside the channel phase,
(ii) cylinder filling with fresh charge,

(iii) exhaust gas expulsion,
(iv) fresh charge channel filling phase.
Explained by Heisler [3], during the 1st phase the exhaust gases are continuously delivered
towards the HPG port where they enter the rotor cells and pass through the channels inducing
their kinetic energy to the fresh char ge. This process generates a pressure wave that is travelling
with the speed of sound inside the fresh charge column till it reaches the rotor cell end. In the 2nd
phase of the PWS operation cycle, the pressure waves compress and accelerate the fresh charg e
trapped into the cells and simultaneously, the far end of the pressurized channels will align to the
outlet port HPA, thus ensuring the delivery of the high pressure charge into the engine’s intake
manifold. Therefore, the engine cylinder is filled with pressurized fresh charge and the port is
closed before the exhaust gases to flow out into the intake manifold. Thus, in the 3rd phase , with
further angular movement of the rotor, the cells containing the trapped exhaust gases get aligned
to the exhaust gas outlet port LPG permitting thus to the exhaust gases to expand furthermore
and to flow out into the exhaust system. This high speed discharge movement of the exhaust
gases creates a vacuum inside the channels they just left and, as the rotor align these c ells to the
fresh charge inlet port LPA, the fresh charge being at atmospheric pressure will rush into the
under -pressurized cells. This is the 4th phase of the PWS operating cycle, when the rotor cells
exposed to the charge inlet port are filled with fresh charge. The vacuum push the charge towards
the exhaust gas exit port, ensuring a useful scavenging of the rotor cells and a cooling effect. The
rotor, continuing to rotate, will align again to the exhaust gas inlet port and, as the cells are now
filled with fresh charge, the cycle of events is repeating, providing thus the engine’s
supercharging.

Fig. – Fluid motion within PWS

The Comprex PWS operat ing principles, looking from the wave propagation perspective, was
thoroughly described by Doerfler, P.K. of Brown, Bovery & Co. [ 54].
In Fig …. is represented the Comprex mode of operation, which can best be exp lained by unfold ing
the axial rotor channels into the projection plane. The rotation of the rotor is transposed into a

movement from the top to down, and the opening and closing of the ports are represented
accordingly to the schedule determined by the geometrical locatio n of the ports. At first, the rotor
is filled with air at rest. As the HPG port opens at its upper edge a compression wave (1) is
created and it propagates from right to left. The wave front moving faster than the speed of sound
compresses the air, thus, t he air occupies less space and permit the exhaust gases to invade more
the rotor cells because of the considerable pressure difference. The boundary interface between
the two fluids is represented with a dotted line, called also “the particle path”, showin g the path of
the first particle entering the wave rotor with high pressure [ 54] and enclosing high kinetic
energy. As the wave reaches the left side, the HPA port opens to let the compressed air flow
towards the intake manifold. The kinetic energy was par tly converted into pressure, the exhaust
gases are therefore decelerated and a second compression wave (2) propagates from left to right.
As this wave reaches the right end of the channels, the HGP port is closed and, as a consequence,
an expansion wave (3 ) originating from the edge runs back towards the left. The HPA port also
closes before the interface arrives to it, preventing the gases to enter into the engine intake
system. When the third described wave reaches the left side, the whole cell’s’ content is at rest. A
slight mixing happen at the interface between the two working media. The pressure is still high
and it helps scavenging the channels when the LPG wide port opens and the rotor content expand
into the engine exhaust system. An expansion wave (4) is induced and, as it travels through the
air and the gases with the speed of sound, it sets the rotor content in motion. It reaches the left
side at the opening edge of the LPA port and it starts drawing fresh air into the cells. The
decreasing speed caused by the pressure loss at opening of both ends of the rotor create the
expansion waves (5 and 7) that travel to the right, decelerating the air to standstill. The pressure
in the exhaust pipe, higher than the fresh charge pressure, induces rather weak compression
waves (6 and 8) that travel back to the left. When the dotted line arrives at the right cells end, all
the gases has left the rotor, together with some scavenging air. The discharge and the intake
ports are closed, the rotor cells are filled w ith fresh air and the cycle is ready to restart.

Fig – The wave propagation schema

The pressure wave speed depends on the speed of sound, that is function of the fluid's’
temperature. Therefore, a PWS will work properly for a specific exhaust gas temperature and at a
given rotor speed. Therefore, the PWS can operate in a narrow range of speeds and load
performances, because the travel time of the pressure wave need to be correlated to the
peripheral speed, in concordance with the engine load and sp eed [ 3]. The wave rotor operating
range is extended by shaping "pockets" into the fixed rotor ends, as shown in Fig….. The pockets
are used to prevent the reflection of the waves in a closed cell end, a process which would cause
inside the channel a signi ficant change of the flow velocity [ 56]. The pockets, marked KT, ET and
GT in the Fig…, allow the fluid to flow from one cell to adjacent ones. They modify the fluid flow so
that the additional pressure waves that appear will correct the response of the PW S to changes in
engine load or speed. At low peripheral speeds the compression pocket KT prolongs the filling
process and, as the engine speed varies, the pressure curve obtained stays relatively flat. Thus, a
modest increase in air density results. In con trast, the expansion pocket ET and the gas pocket GT
improve the scavenging effect at all speeds [ 3]. Therefore, the device can operate with acceptable
performance at other loads and speeds because the pockets allow the particle paths to change
without maj or losses. [ 55 in 56]

Controlling of the pressure ratio can be realized by wastegate method ( Fig. – constructia rotorului
PWS) , when the exhaust gases bypass the wave rotor when the charge pressure inside the engine
intake manifold has the maximum desired value ( Fig. Efectul… .). This value is transmitted to the
wastegate valve actuator that is gradually opened/closed and thus, a part of the exhaust gases
are released into the engine exhaust system instead of entering into the rotor channels.

Fig. – Efectul turației asupra raportului de presiuni cu sau fără
wastegate (inspirat din Heisler)

The pressure of the air to be induced into the engine cylinder is determined by the strength of the
compression waves, that is, as described above, function of the engine’s exhaust gases
temperature. The mass flow entering and leaving the engine are consid ered equal, the energy
contained within the exhaust gases that expand inside the PWS can be sufficient for rising the
intake air pressure. The surplus power covers the inefficiencies: losses caused by the leakage into
the clearance between the rotor and th e stator, losses caused by the incomplete recovery of the
kinetic energy of the compressed air inside the intake air manifold and also of the kinetic energy
of low -pressure exhaust gases leaving to the engine exhaust system, as well as local pressure
losses caused by passing of the fluid over the ports’ edges, work dissipation due to flow friction on
the channel walls [ 57]. The compression and expansion processes in a PWS are non -adiabatic,
different distinctively from the adiabatic compression and expansion in turbomachines. An overall
combined energy efficiencies of 74% is declared by Berchtold in [ 57] taking into accoun t the total
of all losses, making the Comprex competitive wi th turbosuperchargers .

Pressure wave technology benefits and shortcomings

Compared with the turbo -machines, from the performance and construction point of view, the
pressure exchanger’s advant ages can be outlined in short:
– Robust and reliable structure;
– More erosion resistant due to easier uptake of particles or droplets contained into the
working fluids;
– Better resistance to high temperatures because the exposure to material maximum
temperatur es is rather short and in some cases a cooling effect is ensured by the cold fresh
air flow;
– Higher tolerance to pick transitory pressure values;
– Lower rotational speeds of the rotor than the turbo -machines; allowing a better cell filling
or scavenging;
– Faster response to operational transients;
– High compression at low engine speeds;
– Important potential to generate significant rise of pressure in a short time;
– Comparable values of isentropic efficiencies of the compression and expansion in
comparison with the turbomachines;
– More, ideal shock compression efficiency for an ideal pressure exchanger with no friction
considered, for the same increase of the pressure ratio, can significantly overcome a
compressor or a subsonic diffuser efficiency [ 5]; ????????? ??????????
– No surge limit, as experienced at axial or centrifugal turbo -compressors;
– Lower production costs comparing to some performant equipments.
The pressure wave disadvantages are:
– Significant noise level, caused by the non -steady phenomena that occur inside the rotor
cells and by the rotational speed range of the wave rotor that brings the frequency of the
noise into the audible zone;
– Low mass flow rate considered on the frontal area, compared with the turbomachines axial
flow rate [ 58];
– Problems indu ced by fatigue on areas subject to cyclical fluctuations in pressure.

When using the pressure wave technology for ICE supercharging, the main advantage of a PWS
over the conventional turbocharger lies in its immediate load response, since the transfer of high
exhaust gas energy towards the intake air is instantly realized. A lso, the PWS shows its
operational efficiency at low speeds as well as at high speeds. Furthermore, the PWS has no lag
caused by inertia comparing to the turbochargers. The reduced weight and compact dimensions
reveal the PWS potential to be used in superc harging applications of small engines. While the PWS
efficiency is independent of its dimensions, small engine supercharging (below 75kW) – where
turbochargers are deficient – is very adequate for PWS [9]. Other advantages outlined by ITE
Circuit Breaker Company based on road testing with a diesel truck, are: clean exhaust, fewer
gearshifts, improved fuel consumption, no danger of over -speed and low sens itivity to unbalance
[9].

The cell wave rotor disadvantages that can be particularly outlined are:
– The necessity of mechanical driving of the rotor, even though only 0,5% of the engine
power is used;
– PWS is very sensitive to the changes in the pressure losses inside the intake or exhaust
systems (such as the losses caused by the soot filter clogging);
– The limitation of the start -up functioning of the PWS when exhaust gases can escape into
the intake manifold and prevent the engine operation;

– Manufacturing difficulties that can rise the PWS costs of production and, therefore, the final
price [ 59].
– Limited flexibility in montage the PWS system, due to belt drive;
– Significant sensitivity of PWS operation to increased resistance on the low -pressure side [ 6]
– High quantities of exhaust gas and scavenging air;
– Noisy operation;
– The difficulty of designing the PWS, that include finding the optimal geometrical
configuration, controlling, as well as modelling and understanding of the complex unsteady
phenomena that occur inside the rotor cells, sealing and expansion issues, mechanical
problems, etc.

4. PWS modeling and calculations.
4.1. Modeling and gove rning equations of viscous flow [60]

INTRODUCTION
In order to model the pressure wave compressor processes we need to build the basic equations
based of fundamental physical principles for viscous flow, to choose suitable suitable model of the fluid
and to obtain mathematical equations which properly describe the physics of the flow. In this part the
equations wil be presented for unsteady, compressible and viscous three -dimensiona flow in a general
form. Th e viscous approach was choosen because the effects of viscosity, thermal conduction, and
mass diffusion are important. Also, particularized models for pressure wave compressor used during
the years are presented after the general model. The steps for mathe matical modeling usually are:

I. State three fundamental physic al principles of nature , namely:
a. Mass is conserved (i.e., mass can be neither created nor destroyed).
b. Newton’s second law: force = mass × acceleration.
c. Energy is conserved; it can onl y change from one form to another.

II. Determine a suitable model of the fluid : the finite control volume, the infinitesimal fluid element and
molecular model.

III. Apply the fundamental physical principles to the choosen model of the fluid in order to o btain
mathematical equations which properly describe the flow.

MODELS OF THE FLUID
The Finite Control Volume Model considers a volume of interest in a system. It defines a
control volume , noted CV. The control volume may be fixed in space with the fluid moving through it,
and a finite region of the flow containing a large ammount of molecules that can describe the whole
system. The control surface , (CS), is defined as the closed surface w hich bounds the control volume.
The fundamental principles are applied to the fluid inside the CV, and to the fluid crossing the CS.
The Infinitesimal Fluid Element Model c onsiders an infinitesimally small fluid element in the
flow, with a differential v olume dV. The fluid element is called infinitesimal and it is large enough to
contain a large number of molecules so that it can be viewed as a continuous medium. The differential
calculus can be applyed to this model.
The Molecular Model is considered i f the fluid flow is an approach of the mean motion of
molecules from the flow, a microscopic model of the flow wherein the fundamental laws are applied to

the atoms and molecules. This approach is suitable to statistical averaging modeling to define fluid
properties using kinetic theory.

EQUATIONS FOR AN UNSTEADY, COMPRESSIBLE, THREE -DIMENSIONAL VISCOUS FLOW

MASS CONSERVATION EQUATION
The concept of mass flow. Consider an area A arbitrarily oriented in a flow field as shown in
Figure x.18. Let A be small enough such that the flow velocity V is uniform across A. Consider the
fluid elements with velocity V that pass through A. In time dt after crossing A, they have moved a
distance V dt and have swept out the shaded volume shown in Figure x.18. This volume is equal to the
base area A times the height of the prism Vndt, where Vn is the component of velocity normal to A;
then the volume is, Volume = (Vndt)A.
Fig. x.1…..

The mass inside the shaded volume is therefore
Mass=ρ Vndt A
By definition, the mass flow through A is the mass crossing A per second, noted ṁ:
𝑚̇=𝜌(𝑉𝑛𝑑𝑡)𝐴
𝑑𝑡 𝑜𝑟 𝑚̇=𝜌𝑉𝑛𝐴

An important definition is the mass flux (or mass velocity), defined as the mass flow per unit
area.
Mass flux= ṁ
A =ρVn

The mass flux across a surface is equal to the product of density times the component of
velocity perpendicular to the surface. Many of the equations of fluid flow i nvolve products of density
and velocity. In cartesian coordinates, the components of velocity u, v and w and the products ru, rv
and rw are related to remember that these products are the mass fluxes in the x, y, and z directions,
since the velocity compon ents are normal velocities.
In words the conservation of mass is:

Time rate change of mass
inside CV = Net mass flow out of
CV through CS

THE CONTINUITY EQUATION

The integral form of continuity equation is,
∯𝜌𝑽∙𝑑𝑺
𝑆⏟
𝑁𝑒𝑡 𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑜𝑢𝑡
𝑜𝑓 𝐶𝑉 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝐶𝑆= −𝜕
𝜕𝑡∰𝜌𝑑𝒱
𝒱⏟
𝑇𝑖𝑚𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒
𝑜𝑓 𝑚𝑎𝑠𝑠 𝑖𝑛𝑠𝑖𝑑𝑒 𝐶𝑉

The partial differential equation (PDE) form of continuity equation (called also the conservativ e form):
𝜕𝜌
𝜕𝑡+∇∙(𝜌𝑽)=0
or in extended form
𝜕𝜌
𝜕𝑡+𝜕(𝜌𝑢)
𝜕𝑥+𝜕(𝜌𝑣)
𝜕𝑦+𝜕(𝜌𝑤)
𝜕𝑧=0

The material derivative form of continuity equation (or called nonconservative form):
𝑫𝜌
𝑫𝑡+𝜌∇∙𝑽=0
where: 𝒱 is the control volume, S is control surface, ρ is density , V is the velocity field V=ui+vj+wk,
and the substantial derivative in cartesian coordinates which is:
𝑫
𝑫𝑡≝𝜕
𝜕𝑡+𝑢𝜕
𝜕𝑥+𝑣𝜕
𝜕𝑦+𝑤𝜕
𝜕𝑧
or,
𝑫
𝑫𝑡⏟
𝑠𝑢𝑏𝑠𝑡𝑎𝑛𝑡𝑖𝑎𝑙
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒≝ 𝜕
𝜕𝑡⏟
𝑙𝑜𝑐𝑎𝑙
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒+(𝑽∙∇)⏟
𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑣𝑒
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒

where physically meanings are: for the substantial derivative , which is the time rate of change of a
moving fluid element, the local derivative, which is the time rate of change at a fixed point and the
convective derivative, which is the time rate of change due to the movement of the fluid element from
one station to another station in the flow field considering the flow properties are spatially different.

THE MOMENTUM CONSERVATION EQUATION
The general form of Newton’s second law is
F= d
dx(mV)
where mV is the momentum of a body of mass m and F is the resultant force acting on the body to
reach the equilibrium state. This represents the second fundamental principle on which theoretical fluid
dynamics is developed. In words this principle in nonconservative form is:

Time rate change of
momentum = Force [N]
or

Time rate change of
momentum due to unsteady
changes of flow properties
inside volume V + Net flow of
momentum out of
control volume V
across the surface S = Force acting on the fluid
as it flows through
control volume V (x.x)

The relation is applyed to the model of a finite CV, fixed in space, to obtain expressions for
both the left and right sides of Eq. (x.x) in terms of the flow -field scalar variables p, r and vectorial
variable V.
The expression for force F, which is the force acting on the fluid as it flows through the control
volume, comes from two sources: the body forces, i.e. gravity, electromagnetic forces or any other
forces which acts with no contact on the fluid inside V, and surface forces like pressure and shea r stress
acting on the control surface S.

The integral form of momentum equation

𝜕
𝜕𝑡∰𝜌𝑽𝑑𝒱
𝒱⏟
𝑇𝑖𝑚𝑒 𝑟𝑎𝑡𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑑𝑢𝑒 𝑡𝑜 𝑢𝑛𝑠𝑡𝑒𝑎𝑑𝑦 𝑓𝑙𝑜𝑤 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑖𝑒𝑠 𝑖𝑛𝑠𝑖𝑑𝑒 𝒱+ ∯(𝜌𝑽∙𝑑𝑺) 𝑽
𝑆⏟
𝑁𝑒𝑡 𝑓𝑙𝑜𝑤 𝑜𝑓 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑜𝑢𝑡 𝑜𝑓 𝐶𝑉 𝑎𝑐𝑟𝑜𝑠𝑠 𝐶𝑆=
=−∯𝑝𝑑𝑺
𝑆⏟
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑓𝑜𝑟𝑐𝑒 +∰𝜌𝒇𝑑𝒱
𝒱⏟
𝐵𝑜𝑑𝑦 𝑓𝑜𝑟𝑐𝑒 +𝑭𝑣𝑖𝑠𝑐⏟
𝑉𝑖𝑠𝑐𝑜𝑢𝑠
𝑓𝑜𝑟𝑐𝑒

PDE form of momentum equation, conservative form:

𝜕(𝜌𝑢)
𝜕𝑡+∇∙(𝜌𝑢𝑽)=−𝜕𝑝
𝜕𝑥+𝜌𝑓𝑥+(𝔽𝑥)𝑣𝑖𝑠𝑐
𝜕(𝜌𝑣)
𝜕𝑡+∇∙(𝜌𝑣𝑽)=−𝜕𝑝
𝜕𝑦+𝜌𝑓𝑦+(𝔽𝑦)𝑣𝑖𝑠𝑐
𝜕(𝜌𝑤)
𝜕𝑡+∇∙(𝜌𝑤𝑽)=−𝜕𝑝
𝜕𝑧+𝜌𝑓𝑧+(𝔽𝑧)𝑣𝑖𝑠𝑐

Material derivative form of momentum equation, nonconservative form:

𝜌𝑫𝑢
𝑫𝑡=−𝜕𝑝
𝜕𝑥+𝜌𝑓𝑥+(𝔽𝑥)𝑣𝑖𝑠𝑐
𝜌𝑫𝒗
𝑫𝑡=−𝜕𝑝
𝜕𝑦+𝜌𝑓𝑦+(𝔽𝑦)𝑣𝑖𝑠𝑐
𝜌𝑫𝑤
𝑫𝑡=−𝜕𝑝
𝜕𝑧+𝜌𝑓𝑧+(𝔽𝑧)𝑣𝑖𝑠𝑐

Or in vectorial form:

𝜌𝑫𝑽
𝑫𝑡=−∇𝑝+𝜌𝒇+(𝔽)𝑣𝑖𝑠𝑐

where (𝜌𝑓𝑥), (𝜌𝑓𝑦) , (𝜌𝑓𝑧) are the volumetric body force components and (𝔽𝑥)𝑣𝑖𝑠𝑐, (𝔽𝑦)𝑣𝑖𝑠𝑐, (𝔽𝑧)𝑣𝑖𝑠𝑐
are cartezian components of volumetric viscous forces.

To develop the components of pressure and of viscous forces the Figure x.2. is represented only
for x direction. After the equilibrium on x direction regarding the elementar three dimensional elem ent
(Figure x.2) the volumetric viscous forces are:

(𝔽𝑥)𝑣𝑖𝑠𝑐= 𝜕(𝜏𝑥𝑥)
𝜕𝑥+𝜕(𝜏𝑦𝑥)
𝜕𝑦+𝜕(𝜏𝑧𝑥)
𝜕𝑧
(𝔽𝑦)𝑣𝑖𝑠𝑐= 𝜕(𝜏𝑥𝑦)
𝜕𝑥+𝜕(𝜏𝑦𝑦)
𝜕𝑦+𝜕(𝜏𝑧𝑦)
𝜕𝑧
(𝔽𝑧)𝑣𝑖𝑠𝑐=𝜕(𝜏𝑥𝑧)
𝜕𝑥+𝜕(𝜏𝑦𝑧)
𝜕𝑦+𝜕(𝜏𝑧𝑧)
𝜕𝑧

in function of normal stress 𝜏𝑥𝑥 , 𝜏𝑦𝑦 , 𝜏𝑧𝑧 and tangential stresses 𝜏𝑥𝑦 , 𝜏𝑦𝑥 , 𝜏𝑦𝑧 , 𝜏𝑧𝑦 , 𝜏𝑥𝑧 and 𝜏𝑧𝑥.
These components will be expanded later.

Figure x.2 Infinitesimally moving fluid element for volumetric forces.

The vectorial form of volumetric viscous forces using the viscous stress tensor is:

(𝔽̅)𝑣𝑖𝑠𝑐=∇̅∙𝜏̿=[𝜕
𝜕𝑥 𝜕
𝜕𝑦 𝜕
𝜕𝑧]{𝜏𝑥𝑥𝜏𝑥𝑦𝜏𝑥𝑧
𝜏𝑦𝑥𝜏𝑦𝑦𝜏𝑦𝑧
𝜏𝑧𝑥𝜏𝑧𝑦𝜏𝑧𝑧}=
[ 𝜕(𝜏𝑥𝑥)
𝜕𝑥+𝜕(𝜏𝑦𝑥)
𝜕𝑦+𝜕(𝜏𝑧𝑥)
𝜕𝑧
𝜕(𝜏𝑥𝑦)
𝜕𝑥+𝜕(𝜏𝑦𝑦)
𝜕𝑦+𝜕(𝜏𝑧𝑦)
𝜕𝑧
𝜕(𝜏𝑥𝑧)
𝜕𝑥+𝜕(𝜏𝑦𝑧)
𝜕𝑦+𝜕(𝜏𝑧𝑧)
𝜕𝑧] 𝑇
𝑰̅
THE ENERGY EQUATION
For a compressible and viscous flow, the density r is an additional variable, and is therefore
need an additional fundamental equation to complete the system of equations, that is usually the
equation of state for the fluid. The fundamental relation for the energy equation, added two additional
flow-field variables: the internal energy e and temperature T.
The physical statement of the first law of thermodynamics for control volume(CV) is:

The change of
internal energy = Heat transferred from
surroundings + Work done by the
surroundings
or
Time rate of change of
energy of flow as it flows
through control volume V + Rate of heat
transferred from
surroundings to
control volume V = Rate of work done
on fluid inside
control volume V (x.x)

The integral form of energy equation
𝜕
𝜕𝑡∰𝜌(𝑒+𝑽2
2)𝑑𝒱
𝒱
⏟
𝑇𝑖𝑚𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒
𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑠𝑖𝑑𝑒 𝐶𝑉
𝑑𝑢𝑒 𝑡𝑜 𝑢𝑛𝑠𝑡𝑒𝑎𝑑𝑦 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛𝑠
𝑜𝑓 𝑓𝑙𝑜𝑤 𝑓𝑖𝑒𝑙𝑑 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠+∯𝜌(𝑒+𝑽2
2)𝑽∙𝑑𝐒=
𝑆⏟
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦
𝑜𝑓 𝑓𝑙𝑜𝑤 𝑎𝑐𝑟𝑜𝑠𝑠 𝐶𝑆∰𝜌𝒒̇ 𝑑𝒱
𝒱+
⏟
𝑅𝑎𝑡𝑒 𝑜𝑓
𝑣𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 ℎ𝑒𝑎𝑡𝑖𝑛𝑔
+ 𝑄̇𝑣𝑖𝑠𝑐⏟
𝑅𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡
𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛 𝑡𝑜 𝐶𝑉
𝑑𝑢𝑒 𝑡𝑜 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑒𝑓𝑓𝑒𝑐𝑡𝑠−∯𝑝𝑽∙𝑑𝑺+
𝑆⏟
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑤𝑜𝑟𝑘
𝑑𝑜𝑛𝑒 𝑜𝑛 𝑓𝑙𝑢𝑖𝑑 𝑖𝑛𝑠𝑖𝑑𝑒 𝐶𝑉
𝑑𝑢𝑒 𝑡𝑜 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑓𝑜𝑟𝑐𝑒 𝑜𝑛 𝐶𝑆

+∰𝜌(𝒇∙𝑽)𝑑𝒱
𝒱⏟
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑤𝑜𝑟𝑘
𝑑𝑜𝑛𝑒 𝑜𝑛 𝑓𝑙𝑢𝑖𝑑 𝑖𝑛𝑠𝑖𝑑𝑒 𝐶𝑉
𝑑𝑢𝑒 𝑡𝑜 𝑏𝑜𝑑𝑦 𝑓𝑜𝑟𝑐𝑒𝑠 + 𝑊̇𝑣𝑖𝑠𝑐⏟
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑤𝑜𝑟𝑘
𝑑𝑜𝑛𝑒 𝑜𝑛 𝑓𝑙𝑢𝑖𝑑 𝑑𝑢𝑒
𝑡𝑜 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑒𝑓𝑓𝑒𝑐𝑡𝑠 𝑜𝑛 𝐶𝑆

PDE form of energy equation, conservative form:

𝜕
𝜕𝑡[𝜌(𝑒+𝑽2
2)]+∇∙[𝜌(𝑒+𝑽2
2)𝑽]=𝜌𝒒̇+∇∙𝒒̇𝑣𝑖𝑠𝑐−∇∙(𝑝𝑽)+𝜌(𝒇∙𝑽)+𝒘̇𝑣𝑖𝑠𝑐
or in terms of enthalpy:
𝜕
𝜕𝑡[𝜌(ℎ+𝑽2
2)]−𝜕𝑝
𝜕𝑡+∇∙[𝜌(ℎ+𝑽2
2)𝑽]=𝜌𝒒̇+∇∙𝒒̇𝑣𝑖𝑠𝑐+𝜌(𝒇∙𝑽)+𝒘̇𝑣𝑖𝑠𝑐
where the enthalpy ℎ=𝑒+𝑝
𝜌
Material derivative form of energy equation, nonconservative form:

𝜌𝑫
𝑫𝑡(𝑒+𝑽2
2)=𝜌𝒒̇+∇∙𝒒̇𝑣𝑖𝑠𝑐−∇∙(𝑝𝑽)+𝜌(𝒇∙𝑽)+𝒘̇𝑣𝑖𝑠𝑐
where:
 e is the specific internal energy, i.e. for ideal gas is 𝑒= 𝑐𝑣𝑻= 𝑹
𝜸−𝟏∙𝑻=𝟏
𝜸−𝟏∙𝒑
𝝆
 (𝒒̇) is the rate heat addition per unit mass (includes absorption or emission of radiation),

 (∇∙𝒒̇𝑣𝑖𝑠𝑐) is the volumetric rate heating due to viscous effects (includes heat transfer across the
surface due to temperature gradients, i. e., thermal conduction, thermal diffusion; the heating of
fluid element only by thermal conduction is
∇∙𝒒̇𝑣𝑖𝑠𝑐= − (𝜕𝑞̇𝑥
𝜕𝑥+𝜕𝑞̇𝑦
𝜕𝑦+𝜕𝑞̇𝑧
𝜕𝑧 )
or using Fourier law for heat transfered by conduction across normal elementar surface, i.e. 𝒒̇𝑥=
−𝑘 𝜕𝑻
𝜕𝑥 for x direction; so the form of total heat transfered by conduction is
∇∙𝒒̇𝑣𝑖𝑠𝑐=∇∙𝒒̇𝑐𝑜𝑛𝑑=𝜕
𝜕𝑥(𝑘 𝜕𝑻
𝜕𝑥)+ 𝜕
𝜕𝑦(𝑘 𝜕𝑻
𝜕𝑦)+ 𝜕
𝜕𝑥(𝑘 𝜕𝑻
𝜕𝑧)=∇∙(𝑘∇𝑻)

 [−∇∙(𝑝𝑽)] is the volumetric rate work done on fluid by pressure force, which is
−∇∙(𝑝𝑽)=− (𝜕(𝑝𝑢)
𝜕𝑥+𝜕(𝑝𝑣)
𝜕𝑦+𝜕(𝑝𝑤)
𝜕𝑧 )

 (𝜌(𝒇∙𝑽)) is volumetric rate work done on fluid by body forces (gravity field, electromagnetic
field) or any field which acts with no contact on fluid element)

To develop the components of viscous work in the Figure x.x. the rate of heat and work is
represented only for x direction.

Figure x.3 Infinitesimally moving fluid element for volumetric viscous work and heat.

𝒘̇𝑣𝑖𝑠𝑐 is the volumetric work done by the shear stresses, which on elementar fluid element is done on
faces normal to x direction that is
𝒘̇𝑣𝑖𝑠𝑐,𝑥= 𝜕(𝑢𝜏𝑥𝑥)
𝜕𝑥+𝜕(𝑣𝜏𝑦𝑥)
𝜕𝑦+𝜕(𝑤𝜏𝑧𝑥)
𝜕𝑧
in function of normal stress 𝜏𝑥𝑥 and tangential stresses 𝜏𝑦𝑥 and 𝜏𝑧𝑥 .
Then ,the total volumetric viscous work is the sum of all components, that is:
𝒘̇𝑣𝑖𝑠𝑐= 𝜕(𝑢𝜏𝑥𝑥)
𝜕𝑥+𝜕(𝑢𝜏𝑦𝑥)
𝜕𝑦+𝜕(𝑢𝜏𝑧𝑥)
𝜕𝑧+
+ 𝜕(𝑣𝜏𝑥𝑦)
𝜕𝑥+𝜕(𝑣𝜏𝑦𝑦)
𝜕𝑦+𝜕(𝑣𝜏𝑧𝑦)
𝜕𝑧+ 𝜕(𝑤𝜏𝑥𝑧)
𝜕𝑥+𝜕(𝑤𝜏𝑦𝑧)
𝜕𝑦+𝜕(𝑤𝜏𝑧𝑧)
𝜕𝑧 =𝜏∶∇𝑽

VISCOSITY, THERMAL CONDUCTION and SHEAR STRESS
The viscosity and thermal conduction in a fluid are due to the transport of momentum and
energy by random molecular motion.
The transport of molecular momentum gives rise to the macroscopic effect which is the
viscosity, and the transport of molecular energy gives rise to the macroscopic effect which is thermal
conduction.
Consider the flow presented in Figure X.4, a one dime nsional flow with horizontal streamlines
in the x direction but with velocity and temperature gradients in the y direction of velocity.

Figure x.4. Shear stress and heat conduction for unidimensional flow.

The shear stress acting on a plane perpendicular to the y axis is denoted by τyx and is
proportional to the velocity gradient in the y direction, by a constant of proportionality. This constant is
named as the viscosity coefficient μ, and defined as 𝜇≝𝜏𝑦𝑥
(𝜕𝑢
𝜕𝑦), then shear stress is:
𝜏𝑦𝑥=𝜇𝜕𝑢
𝜕𝑦
In addition, the time rate of heat conducted per unit area across plane ab in Figure X.XX is
denoted by q̇y and is proportional to the temperature gradient in the y direction, and a constant of
proportionality which is defined as the thermal conductivity k:
𝑞̇𝑦=−𝑘𝜕𝑇
𝜕𝑦

Both μ and k are physical properties of the fluid and are functions of temperature on ly. A
conventional used relation for moderate temperature variation of μ for air is given by Sutherland’s
law,
𝜇
𝜇0=(𝑇
𝑇0)1.5𝑇0+𝑆
𝑇+𝑆

where T is temperature in Kelvin and 𝜇0 is a reference viscosity at a reference temperature T0 and S is a
constant , S = 110.98 K. In the normal standard condition for air the refernce viscosity 𝜇0 =1.789410-5
kg/ms and reference temperature 𝑇0 = 288.15 K.
The temperature variation of thermal conductivity k is for air at standard conditions,
𝑘=1.45∙𝜇∙𝑐𝑝
where cp = 1000 J/(kg K).

In fluid mechanics the shear stress is proportional to the time rate of strain. The time rate of
strain for elementar volume of fluid in the ( xy), (yz) and ( xz) plane are:

𝜀𝑥𝑦=𝜕𝑣
𝜕𝑥+𝜕𝑢
𝜕𝑦 ,𝜀𝑦𝑧=𝜕𝑤
𝜕𝑦+𝜕𝑣
𝜕𝑧 ,𝜀𝑥𝑧=𝜕𝑤
𝜕𝑥+𝜕𝑢
𝜕𝑧

Since tangential stress is proportional to the shear stress and assume the moment generated by
the tangential stresses in the fluid element between two parallel faces is zero, the expression of
tangential stress is:
𝜏𝑥𝑦= 𝜏𝑦𝑥=𝜇(𝜕𝑣
𝜕𝑥+𝜕𝑢
𝜕𝑦)
and
𝜏𝑦𝑧= 𝜏𝑧𝑦=𝜇(𝜕𝑤
𝜕𝑦+𝜕𝑣
𝜕𝑧)
𝜏𝑥𝑧= 𝜏𝑧𝑥=𝜇(𝜕𝑢
𝜕𝑧+𝜕𝑤
𝜕𝑥)

The normal stresses 𝜏𝑥𝑥 , 𝜏𝑦𝑦 and 𝜏𝑧𝑧 which appears in the case of viscous flow and acting in
addition to the pressure as force on normal to a surface.
These normal stresses act to the fluid element compressing or expanding it, hence the element is
changing its volume. Using the physically interpretation of divergence of velocity related to time rate of
change of elementar volume per volume of the element, and the derivatives (𝜕𝑢
𝜕𝑥 , 𝜕𝑣
𝜕𝑦 ,𝜕𝑤
𝜕𝑧) are related
to deformation of the element to ( ∇V).
The expressions of the normal stresses are:
𝜏𝑥𝑥=2𝜇𝜕𝑢
𝜕𝑥+𝜆(𝛁∙𝑽)
𝜏𝑦𝑦=2𝜇𝜕𝑣
𝜕𝑦+𝜆(𝛁∙𝑽)
𝜏𝑧𝑧=2𝜇𝜕𝑤
𝜕𝑧+𝜆(𝛁∙𝑽)

In the above equations  is called the bulk viscosity coefficient , or called as the second viscosity
coefficient . The Stokes hypothesis for the second viscosity coefficient is λ=−2
3μ .
For incopressible flow λ is irrelevant, but the correct expression for the bulk viscosity for
compressible flows is still difficult to measure. The normal stresses are important only in that cases
where the derivatives (𝜕𝑢
𝜕𝑥 , 𝜕𝑣
𝜕𝑦 ,𝜕𝑤
𝜕𝑧) are very large. For many pr actical flow problems the normal
stresses 𝜏𝑥𝑥 , 𝜏𝑦𝑦 and 𝜏𝑧𝑧 are small. In the case of shock waves regarding the internal structure of the
shock wave layer this is finite and with a small thickness of normal layer. For a normal shock wave
across wh ich large changes in velocity appair over a tiny distance, the layer is typically 10-7 m
[Anderson, 2011, 2001, 1991], then 𝜕𝑢
𝜕𝑥 is very large and the normal stress 𝜏𝑥𝑥 becomes important inside
the shock wave.
Substituting normal, tangential stres ses in the expression of viscous froces the expression of
these components are:

(𝔽𝑥)𝑣𝑖𝑠𝑐= 𝜕
𝜕𝑥(2𝜇𝜕𝑢
𝜕𝑥+𝜆(𝛁∙𝑽))+𝜕
𝜕𝑦[𝜇(𝜕𝑣
𝜕𝑥+𝜕𝑢
𝜕𝑦)]+𝜕
𝜕𝑧[𝜇(𝜕𝑢
𝜕𝑧+𝜕𝑤
𝜕𝑥)]
(𝔽y)visc=∂
∂x[μ(∂v
∂x+∂u
∂y)]+ ∂
∂y(2μ∂v
∂y+𝜆(𝛁∙𝑽))+∂
∂z[μ(∂w
∂y+∂v
∂z)]

(𝔽z)visc= ∂
∂x[μ(∂u
∂z+∂w
∂x)]+∂
∂y[μ(∂w
∂y+∂v
∂z)]+∂
∂z(2μ∂w
∂z+𝜆(𝛁∙𝑽))

An expresion of viscous stress tensor in vectorial maner is:
(𝔽̅)𝑣𝑖𝑠𝑐=(𝔽𝑥)𝑣𝑖𝑠𝑐𝒊̅+(𝔽𝑦)𝑣𝑖𝑠𝑐𝒋̅+(𝔽𝑧)𝑣𝑖𝑠𝑐𝒌̅=∇̅∙𝜏̿
𝜏̿=𝜇(∇𝑽+ ∇𝑽𝑻)+𝜆(∇∙𝐕)𝑰
In expanded form the stress tensor is:
𝜏̿=𝜇
{ 𝜕𝑢
𝜕𝑥+𝜕𝑢
𝜕𝑥𝜕𝑣
𝜕𝑥+𝜕𝑢
𝜕𝑦𝜕𝑢
𝜕𝑧+𝜕𝑤
𝜕𝑥
∂v
∂x+∂u
∂y∂v
∂y+∂v
∂y∂w
∂y+∂v
∂z
∂u
∂z+∂w
∂x∂w
∂y+∂v
∂z∂w
∂z+∂w
∂z}
+
+𝜆
{ 𝜕𝑢
𝜕𝑥+𝜕𝑣
𝜕𝑦+𝜕𝑤
𝜕𝑧0 0
0𝜕𝑢
𝜕𝑥+𝜕𝑣
𝜕𝑦+𝜕𝑤
𝜕𝑧0
0 0𝜕𝑢
𝜕𝑥+𝜕𝑣
𝜕𝑦+𝜕𝑤
𝜕𝑧}

GENERAL FORM OF EQUATIONS
The PDE equations presented below represent the complete Navier -Stokes equations for an
unsteady, compressible, three -dimensional viscous flow. The mass species conservation, the
momentum diffusion of species and thermal diffusion are not considered in this approach.
PDE form of continuity equation:

𝜕𝜌
𝜕𝑡+𝜕(𝜌𝑢)
𝜕𝑥+𝜕(𝜌𝑣)
𝜕𝑦+𝜕(𝜌𝑤)
𝜕𝑧=0

PDE form of momentum equation:

𝜌𝜕𝑢
𝜕𝑡+𝜌𝑢𝜕𝑢
𝜕𝑥+𝜌𝑣𝜕𝑢
𝜕𝑦+𝜌𝑤𝜕𝑢
𝜕𝑧=

=−𝜕𝑝
𝜕𝑥+𝜌𝑓𝑥+ 𝜕
𝜕𝑥(2𝜇𝜕𝑢
𝜕𝑥+𝜆(𝛁∙𝑽))+𝜕
𝜕𝑦[𝜇(𝜕𝑣
𝜕𝑥+𝜕𝑢
𝜕𝑦)]+𝜕
𝜕𝑧[𝜇(𝜕𝑢
𝜕𝑧+𝜕𝑤
𝜕𝑥)]
𝜌𝜕𝑣
𝜕𝑡+𝜌𝑢𝜕𝑣
𝜕𝑥+𝜌𝑣𝜕𝑢
𝜕𝑦+𝜌𝑤𝜕𝑣
𝜕𝑧=
=−𝜕𝑝
𝜕𝑦+𝜌𝑓𝑦+∂
∂x[μ(∂v
∂x+∂u
∂y)]+ ∂
∂y(2μ∂v
∂y+𝜆(𝛁∙𝑽))+∂
∂z[μ(∂w
∂y+∂v
∂z)]
𝜌𝜕𝑤
𝜕𝑡+𝜌𝑢𝜕𝑤
𝜕𝑥+𝜌𝑣𝜕𝑤
𝜕𝑦+𝜌𝑤𝜕𝑤
𝜕𝑧=
=−𝜕𝑝
𝜕𝑧+𝜌𝑓𝑧+∂
∂x[μ(∂u
∂z+∂w
∂x)]+∂
∂y[μ(∂w
∂y+∂v
∂z)]+∂
∂z(2μ∂w
∂z+𝜆(𝛁∙𝑽))

PDE form of energy equation:

𝜕
𝜕𝑡[𝜌(𝑒+𝑽2
2)]+∇∙[𝜌(𝑒+𝑽2
2)𝑽]=
= 𝜌𝒒̇+𝜕
𝜕𝑥(𝑘 𝜕𝑻
𝜕𝑥)+ 𝜕
𝜕𝑦(𝑘 𝜕𝑻
𝜕𝑦)+ 𝜕
𝜕𝑥(𝑘 𝜕𝑻
𝜕𝑧)−(𝜕(𝑝𝑢)
𝜕𝑥+𝜕(𝑝𝑣)
𝜕𝑦+𝜕(𝑝𝑤)
𝜕𝑧 )
+𝜌(𝑢𝑓𝑥+𝑣𝑓𝑦+𝑤𝑓𝑧)+𝜕(𝑢𝜏𝑥𝑥)
𝜕𝑥+𝜕(𝑢𝜏𝑦𝑥)
𝜕𝑦
+𝜕(𝑢𝜏𝑧𝑥)
𝜕𝑧+ 𝜕(𝑣𝜏𝑥𝑦)
𝜕𝑥+𝜕(𝑣𝜏𝑦𝑦)
𝜕𝑦+𝜕(𝑣𝜏𝑧𝑦)
𝜕𝑧+ 𝜕(𝑤𝜏𝑥𝑧)
𝜕𝑥+𝜕(𝑤𝜏𝑦𝑧)
𝜕𝑦+𝜕(𝑤𝜏𝑧𝑧)
𝜕𝑧
considering the expressions of normal and tangential stresses presented above.

The general equation of energy considering the enthalpy of the fluid is:

𝜕
𝜕𝑡[𝜌(ℎ+𝑽2
2)]−𝜕𝑝
𝜕𝑡+∇∙[𝜌(ℎ+𝑽2
2)𝑽]=
= 𝜌𝒒̇+𝜕
𝜕𝑥(𝑘 𝜕𝑻
𝜕𝑥)+ 𝜕
𝜕𝑦(𝑘 𝜕𝑻
𝜕𝑦)+ 𝜕
𝜕𝑥(𝑘 𝜕𝑻
𝜕𝑧)+𝜌(𝑢𝑓𝑥+𝑣𝑓𝑦+𝑤𝑓𝑧)+
+𝜕(𝑢𝜏𝑥𝑥)
𝜕𝑥+𝜕(𝑢𝜏𝑦𝑥)
𝜕𝑦+𝜕(𝑢𝜏𝑧𝑥)
𝜕𝑧+ 𝜕(𝑣𝜏𝑥𝑦)
𝜕𝑥+𝜕(𝑣𝜏𝑦𝑦)
𝜕𝑦+𝜕(𝑣𝜏𝑧𝑦)
𝜕𝑧+
+ 𝜕(𝑤𝜏𝑥𝑧)
𝜕𝑥+𝜕(𝑤𝜏𝑦𝑧)
𝜕𝑦+𝜕(𝑤𝜏𝑧𝑧)
𝜕𝑧

THE EQUATIONS OF COMPRESSIBLE VISCOUS FLOW [61, 62]
A complete set of equations presented in vectorial form are the compressible Navier -Stokes
equations for an isotropic Newtonian fluid with variable properties. First three equations are
conservation of mass, momentum and energy, and the rest of equations are constitutive eq uations to
complete the model.

Dρ
Dt+ρ∇∙V=0 (1 eq.)
ρDV
Dt=−∇p+ρf+∇∙τ (3 eqs.)
ρD
Dt(e+1
2𝑽∙𝑽)=ρq̇gen−∇∙q̇visc−∇∙(pV)+ρ(f∙V)+∇∙(𝜏∙𝑽) (1 eq.)
𝜏=μ(∇V+ ∇VT)+λ(∇∙V)I (6 eqs.)
𝒒̇𝑣𝑖𝑠𝑐=𝒒̇𝑐𝑜𝑛𝑑=−𝑘∇𝑻 (3 eqs.)
𝒒̇𝑔𝑒𝑛=0 (0 eq., neglected)
𝜇=𝜇(𝜌,𝑻) (1 eq.)
𝜆=𝜆(𝜌,𝑻) (1 eq.)
𝑘=𝑘(𝜌,𝑻) (1 eq.)
𝑝=𝑝(𝜌,𝑻) (1 eq.)
𝑒=𝑒(𝜌,𝑻) (1 eq.)

The unknown variables of the above differential system are :
• 𝜌 –density, scalar, 1 variable [ kg/m3]
• V –velocity, vector, 3 variables [ m/s]
• p –pressure, scalar, 1 variable [ N/m2]
• e –internal energy, scalar, 1 variable [ J/kg]
• T –temperature, scalar, 1 variable [ K]
• 𝜏 –viscous stress, symmetric tensor, 6 variables [ N/m2]
• qcond –heat flux vector, vector, 3 variables [ W/m2]
• 𝜇 –first coefficient of viscosity, scalar, 1 variable [ N.s/m2 or kg/m.s ]
• 𝜆 –second coefficient of viscosity, scalar, 1 variable [ kg/m.s ]
• k–thermal conductivity, scalar, 1 variable [ W/m.K ]

The volumetric body force considered in most models is given only by gravitational field and
there f = g and is the constant gravitational acceleration. The result is a differential system with 19
equations and 19 variables which can be solved.
This system must be consistent with considering the second law of thermody namics. The
definition of the entropy s with the Gibbs relation is Tds=de+p d(1
ρ) and the second law of
thermodynamics states:
𝜌𝐷𝑠
𝐷𝑡≥−∇∙(𝒒̇𝐡𝐭
𝑇)
To determine time rate of entropylet the time rate the equation of Gibbs, that is:
T𝐷s
𝐷t=𝐷e
𝐷t+p𝐷
𝐷t(1
ρ)=𝐷e
𝐷t−𝑝
𝜌2Dρ
Dt ⟹𝜌𝐷e
𝐷t=ρT𝐷s
𝐷t+𝑝
𝜌Dρ
Dt
and also using energy ecuation
ρDe
Dt=−∇∙q̇ht−p∇∙𝑽+(𝜏∶∇𝑽) and mass conservation 1
𝜌Dρ
Dt=−∇∙𝑽

then the entropy change is:
𝜌𝐷𝑠
𝐷𝑡=−1
𝑇∇∙q̇ht+1
𝑇(𝜏∶∇𝑽)
As it can be observed the change of entropy is duet o heat transfer and viscous work. Also the
term ∇∙(𝒒̇𝐡𝐭
𝑇)=1
𝑇∇∙𝑞̇ℎ𝑡−𝑞̇ℎ𝑡
𝑇2∇𝑇 so the ecuation for entropy change is:
𝜌𝐷𝑠
𝐷𝑡=−∇∙(𝒒̇ℎ𝑡
𝑇)−𝒒̇ℎ𝑡
𝑇2∇𝑇+1
𝑇(𝜏∶∇𝑽)
and comparing with the second law of thermodynamics, the result is:
−𝒒̇ℎ𝑡
𝑇2∇𝑇+1
𝑇(𝜏∶∇𝑽)≥0
or using only heat transfer by conduction
𝑘∇𝑇∙∇𝑇
𝑇2+1
𝑇(𝜏∶∇𝑽)≥0
The constitutive equations for thermal conductivity 𝑘(𝜌,𝑻), or heat flux 𝑞̇ℎ𝑡 , first viscosity
coefficient 𝜇(𝜌,𝑻), the second viscosity coefficient 𝜆(𝜌,𝑻) and then the viscous stress 𝝉 must be
constructed as that to respect the second law of thermodynamics.

5. Experimental results (nu cred ca e bun subtitlul…)

Performance features. Comparison TC – PWS – NA. Results.

A benchmarking between PWS (pressure wave superchargers), TC (turbochargers), C (compressors) or
NA (naturally aspirated engines) reveal significant differences in terms of efficiency, supercharging
pressure ratios, power, torque, etc. For instance, the Co mprex supercharger register a maximum
adiabatic efficiency of 90%, while a conventional TC hardly reaches 56% (74%???) [3] in normal
operating conditions and a Roots C supercharger even less. The high efficiency of PWS comes from the
internal cooling effec t of fresh air sweeping the channels, while in the TC or C the rise in heat lowers
the adiabatic efficiency.
Comprex is experiencing losses caused by the bearings friction and windage, but they are neglectable.
Also, the power lost by driving the rotor is very low, about 0,5 % of total load [63], and no work is lost
for air compression, as in the case of positive displacement compressors.

As shown in Figs. … and …, a comparison of the power and torque curves of turbochargers, pressure
wave chargers , naturally aspirated engine and mechanical superchargers offset against the basic engine
speed reveal that the TC does not reach the values of the PWS and the C. The mechanical supercharger
gives an approximately equal torque curve over the entire rotationa l speed range as in the case of a
large aspirated motor [64], similar with the NA engine . When comparing the TC with the PWS , it is
obvious that although a very high torque was achieved at 2000 rpm, the maximum power is not
sufficient .

Fig.. Fig..

??? ?????? Inspirate din
Heisler , dar diferite de cele 2 de mai sus, care sunt din lucrarea lui Weidemann (Volkswagen) despre
Comprex ……

In the matter of fuel specific consumption, the mechanichal charger is more effective, in particular at
low loads in comparison to charging systems usin g the exhaust gas energy i.e. turbocharging or
pressure wave supercharging. Therefore, the engine should be operated in bypass mode at lower loads
and only in the range of intermediate pressure of 5 – 6.5 bar should be charged in order to rema in in the
lower consumption zone. At a medium pressure of 2 bar, the PWS has a higher consumption with about
20g/kWh compared to the mechanical charger [64]. However, at high l oads and real traffic conditions,
the PWS has real consumption advantages. A better consumption for PWS comparing to the
turbocharger is achieved at higher torque and lower speeds [ 3].

Fig. INSPIRATA din Wiedemann

6. Future research targets
……………….

7. Conclusions
The Conclusions part is summarizing the advantages and disadvantages of PW turbocharging. It also highlights the challenges t hat
specialists have to overcome, pointing out aspects to become future preoccupations and research directions in or der to achieve higher
engine efficiency and lower fuel consumption, thus, lower CO2 emissions.

Pressure wave chargers are able to fulfill all main functions in the simplest way and with the least time
delay.

În prezent, costurile de fabricație ale unui sistem de supraalimentare cu Comprex este de aprox. două
ori mai mare decât cu turbosuflantă, ceea ce face ca această clasă de superchargers o propunere
economică pe termen lung în care trebuiesc cântărite atent avantajele și dezavantajele.
Cu toate a ceste dezavantaje, un imbold continuu pentru obținerea de eficiență energetică, progresele de
diminuare a tehnologiei mai vechi, precum și schimbarile pietei au stimulat nou interes pentru
tehnologia rotorului cu unde.

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