BASIC GAS DYNAMICS
EXERCISES 3 EXERCISES 3
117
Ans.
Ans.
3·1. What is the basic differ-:nce between compressible and incompressi- ble fluid flow? What difference does it cause in the physical nature of the compressible flow.
3·2. What is velocity of sound ? Derive an equation for the velocity of sound for a perfect gas.
3·3. Explain the terms Mach number, Mach cone, Mach wedge and Mach angle.
3·4. What is subsonic, supersonic, and hypersonic flows ? 3·5. Explain the terms 'zone of silence' and 'zone of action'.
3·6. Explain the concept of total or stagnation properties.
3·7. D~rive expressions for st,ignation temperature, stagnation pressure, stagnat10n density, and stagnation e::,baipy.
3·8. What is one-dimensiona: ar!iabatic flow? Derive expressions for
dp dp dA
p'
-P-and A for such a flow.3·9. What is isentropic fl0·.: 0 Deduce that in an isentropic flow convergent passage acts as a nozzle fJr ';absonic flow and as a diffuser for supera sonic flow, while a divergent flow pu«3age acts as a diffuser for subsonic flow- and as a nozzle for supersonic flow .
3·10. For a variable area /1Jc<r nassage define the terms 'critical area' and
'throat'. ' ·
3·11. Show that the conc'>',.H1 l" maximum mass flow rate for an air nozzle occurs at _£t=0·528.
p
3·12. Discuss the effv.!
o:
t'l""''ire ratio on flow through (a) conver;;cn: :, :.u,..:, (b) convergent-divergent nozzle.SECTION B
3·13.(a) Show t'1Jr llic cr:an;;c in temperature aero<• an infinitesimal pressure pulse travelling through an ideal gas is
,lT=T(k-l)dV a
118 GAS TURBINE .AND JET AND ROCKET PROfULSION
(b) Sl:ow that for an i~entropic flow
*- [ 2c"+V2(k-1) a
-'\J
k+l(c) The velocity of sound of a gas is found experimentally io be 900 rn/~cc wh,n the gas pressure is 1·8 ),gi'/cmi and the density 0·001 kg/ms. What·
is t;,c value of the specific heat ratio cf tJds gas?
3·14. Air is supplied through a nozzle with an exit arc,; of 10 cm2. A tank . s111'.plies the air.at 10 l<gf/cm" and 2C0°C and the discharge pre,sure is 7·5 kgf/
c,1,2. Assuming no los,, determ.ine the discharge temperature, the discharge velocity, the Mach number, and the mass 11ow.
3· 15. A stream of air flowing in a duct is at a pressure of 2 kgf /cm2 , has a Mach numb~r of 0·6, and llows at the rate of 0·:25 kg/sec. The cross-sectional area of the duct is 1 cm2.
(a) Comp1.1te the stagna !ion temperature of the stre3m.
- (b) Wkt is the maximum percentage reduction area which could be introduced witl:cut reducing tbe flow rate of the stream ?
(c) For the maximum area reduction of part (b), find the velocity and pressure at the minimum area, assuming no friction and no heat transfer.
3·16. Consider a supersonic nozzle construct,d with a ratio of exit of the throat area of 2·0. The nozzle 1s supplied with air at low speed at 7 kgf/cm2 and 40°C. Ttie overall nozzle efficiency from inlet to exit is 90 per cent but the
flow is isentropic up !othe throat,-. '
. :,· ·.·. \
Calculate the pressure, velocity arid the Mach number at exit, and com- pare with corresponding values for isertfmpic flows.
3·17. Air is moving in a pipe witha velocity Of 100 m/sec. The tempe- rature and pressure at one section in the pipe is 40°C and 2kgf/cm2, respectively.
(a) Stagnation pressure (b) Stagnation temperature (c) Stagnation density, and (dJ Stagnation enthalpy.
3·18. Prove that the velocity of sound in a Van der Waal's gas is giYen by
a=J
gkRT( (l-~9f -~f)
3·19. The photogr3phs of a bullet in flight show that at a great distaI1ce from the bullet the total included angle of the wave is 50·3°. The pressure and temperature of the undisturbed air are 1 kgf/cm2 and 25°C, respectively. Cal- culate the velocity of the bullet and the Mach number of the bullet. relative to undisturbed air.
3·20. Air approaches and flows around a body. At the stagnation point the pressure is 1 kgf/cm2 and temperature 2l°C. At this point the static press- ure is 0·7 kgf/cm2. Find the Mach number at this point.
3·21. Show that in a flow from re,ervoir, the maximum velocity that may be reached is given by
x=aoJ k:_
1 for a perfect gas.What are the corresponding values of temperature and ·Mach number?
Interpret.
BASIC GAS DYNAMI03
:i.·22. For a perfect gas, prove that (i) a*2(k+ 1)=2a2+ V2(k-1)
(ii) (p/pi)=(p/pi) k_/k-l)(s-.;1)/R
119
3'·23. A convergent-divergent nozzle is fitted into the'side of a large ves,e! containing a g1s under co.1stant pressure and temperature. If the ratio of specific heats of the g1s is t ·3 :1, calculate, from first principles, the percen·
tage.chrn11c in (a) press•.1r~. (b) absolute temperature betwe~n the i:ese,voir and th~ throat of th;! nozzle under t:1,! m 1xiinum flow .conditions. Neglect friction Rnd assu:1c: ~d iabatic expansion. · ·
[(a)-45:6%; (bJ 12% l REFERE-i'CE~
3 1. Shapiro, A.H .. The Dyl]amics and Thefmodyn;1mics of Compressible F'"id Flow, vol. l, N.Y.,The Ro~Jcf Press Company, 1953.
3·2. Chapman A.J., and Walker; W.F., Introductory Gas Dynamics, Holt, Rinelnrt & Winston lnc., N.Y., 1971.
3·3. Cambel A.B •. and Jennin 5s B.H., Gffs DJnamics, McGraw Hill, N.Y.
1958.
3·4 .. Ow.:z1rek. J.A\. Fundamentals of Gar Dynamics, Scranton Pa;
International Text book, 1964.
3·5. Lipmann H.W,, and Roshko A., Elements of Gas Dynamics, N.Y,
Wiley, 19~7. ·
3·6. Benedict, R.P,, and Steitz, W.G., A Generalized approach to 01ie- dimensional gas dynamics, Tr; AS\1:E .: Series A, Vol. 84, No. I_, January 1962 p. 49.
3·7. Steitz, W.G. and Benedict, R. P., Some Generalizations in One- DimensionrJ/ Conitant Dmsity Fluid Dynmnics, Tr. ASME., vol. 8t, series A, No,.
I, January 196:?, p. 44.
3·8. Goldstein, S., Afoderli Developments 111 Fluid Dynamics, Oxford·
University Press, 1938.
3 9. .Kestin; J. ana .luemba, S.K., One-dimensional High-speed F/011·s, Aircraft Engg, June, 1:953. -
3· 10. Courant,
R.,
and Friedrichs, K.O., 'Supersonic Flow and ,C::fzock Wai,es, Intrrscience Publishers, N.Y., 11948.3·1H Anderson.- LR .. et:;/; Axisymnietric One-dimensional Compressible Flow-tlzedry and Applieation,-Tr. ASME, Sel:ks E, vol. 37, No. 4, Dec. 70, pp.
91-7923. .
4
POSITlVE DISPLACEMENT COMPRESSORS
4·1. INTRODUCTION
There is hardly any product used in our daily lives to which com- pressed afr has not . contributed in ,. ,r. e way. Compressed air is widely used in chemical and petr:, d1emical industries, for trans- mitting power, for conveying solid and powdered materials in pipe lines, in mines, and steel industry. Comprefsors are integral part of gas turbine plants. Compre~sors are used for delivering natural gas through Jong distance pipe lines, compressing the mixture of hydro- gen and nitrogen in ammonia synthesis plant, delivering lime-kiln gas for Solvay process, circulating synthesis gases in processes for the manufacture of primary products of plastics, circulation of carbon monoxide or helium for. cooling the nuclear power plants and liquefying ammonia in large refrigerating plants. In addition to steel industry, where about IO per cent of the total capital cost of the plant comprises compressors, it is also used among instrument manufactures, and for transportation, glass, food, paper, rubber and plastic products. In a nitric acid plant about 10 per cent and in an ammonia synthesis plant about 15 to 20 per cent of total initial cost is of compressors. With regard to such a wide variety of industrial applications of compressors, it can be said that compresied air is only next best to electricity and this makes the study of various types of compressors a very important part of the education of a mechanical engineer.
The function of a compressor is to incnase the pressure of the air or the gas inducted. The term fan is used to describe dynamic air compressors in which this increase in presrnre is less than O· 35 kgf/cm2, -i.e. the density of the air does r,ot change appreciably.
Dynamic compressor~, which increase the pressure of the fluid passir,g through them upto about 3 kgf/cm2 are termed as blou:ers. Compres- sors producing vacuum are called exhausters or air pump,s and those which increase the pressure of the fluid already above atmospheric pressures arc called boo8ters. For the analysis of fans incompressible flow equations can be med while the compressibility effects must be considered in design and analysis of blowers and
[SEC. 4•3J POSITIVE DISPLACEMENT COMPRESSORS 121 compressors. In this chapter only the positive displacement com- pressors are discussed. Dynamic compressor form the subject matter of the next chapter.
,s-:z.
CLASSIFICATION OF AIR COMPRESSORSBasically the compressors can be classified into two types, namely : - 1. Positive displacement type compressors, and
2. Dynamic compre~sors.
Positive displacement type of compressors are those in which successive volumes of air or gas are first confined within a closed space and then the pressure in this space is increased by decreasing its volume. Positive displacement type of compressors include reciprocating, sliding-vane rotary, liquid piston, two or three lobe rotary, and rotary screw compressors. The pressure developed by these compressors is independent of speed while the rate of flow changes with speed. Dynamic compressors are those in which the compression of air or gas is affected by the dynamic action of rotating vanes or impellers. These rotating vanes or impellers impart velocity and pressure to the flowing medium. These include centrifugal and a'llial flow compressors. In dynamic compressors the fluid flow is steady flow through the machine unlike disconti • neous flow of positive displacement type compressors. Fig. 14· I shows
the classification of compressors.
Dvnamic
r
. !
(
I
Centrifugal
Sliding
r
van;
Compressors
Axial flow I
I r
Single rotor
I
I
Liquid 1
.piston
I
Mixed
1
flowPositive Displacement
l
I
r
I
l I
Rotary Reciprocating
I I
r
l
ITwo rotor
I
Lobed (Roots blower)
Screw
1
Fig. 4· 1. Classification of compressors.
4·3. COMPRESSOR EFFICIENCIES
(i) IsMhermaJ dRde?: y. Since for isothermal compression the ,\erk requind to drive the ccmpresor is minimum, it is con-
sidt r< d 2s a ,tr.1;da1d tcw2rds which each de~igner will try to arpczch ; and the rnfriman( e o tl e ccrnprecrnr is given by iso- thermal d1iciency, which is definfed as
122 GAS TURBINE .AND JET AND ROCKET PROPUc,~ION LSEC. Lf·,'.{}
Isothermal efficiency
hothermal work
= t\ctu1l indicated work (4·])
<iiJ Volum~td.c efficiency. One of the effects of clearance volume is to reduce the amount of air which can be m·:ked in during the sucdon stroke The miss of air inducted is (urther reduced by the resi'stance of inlet and exhaust valves. heating- of air due tCl compres~or parts, de. The ratio of the mass of air pi·ising t!Fough the compr~ssor · ~nd the mass ofair which wvu!J completely fill the swept volmne is\ de.fined as volumetric efliciency. The n\·eoll volumetric efficiency is given by
Overall volumetric efficiency
:Mass of air delivered
=M--a-~-s-o-f~. -a~i-r-corresponding to swept volume of L.P. cylinder at F.A.D. conditions 4 2(a}
where free air delivery (F.A.D) is defined as tbe volume of air
· delivered and reduced to intake pressure and temperature. The -capacity of a compressor is usualJy given in term, of F.A.D.
Alternatively, Overall volumetric efficienc1
Volume of free air inhaled
= Swept volume of L.P. q:,inder 4·2 (b}
, When .the volumetric efficiency is calculated in terms of the conditions at ·normal prf"SSUFC, and tern peraturc, abbreviated as N
.T
P. ,.it is called absolute volumetric effi-:iencv.
,Absolute volumetric efficiency
M;i.ss of air delivered
- - - = - -
=Mass of air corresponding to swept volume of LP. cylinder at N T.P.
(4·3) .Alternatively
Absolute volumetric efficiency
Volume of air inhaled at N.TP.
Swept volume of L.P. cylinder
(iii) Mechanical efficiency. The mechanical efficiency of a reciprocating compressor is defined as the rado of indicated power to the brake power of the shaft supplied to the compressor.
Mechanical efficienfy
>~- l~dicated or a-tr h.p.
Shaft h.p.
PO!!ITIVE DISPLACEMENT CO'.\IP_R KSSOR'l 123
4·4. RECIPROCATING COMPRESSOR
fhe compres,ing element ofa reciprocating compressor is-a pisfon in a cylinder, employing basically the same mechanical action as tbc intake and compression strokes of a reciprocating internal combus- tion engine. Fig. 4·2 shows a schematic diagram of such a corn;we,~
sor. During the suction stroke, when the piston travds fr,,rn rop dead centre to bott".lm dead cen,rc, the inlet. valve opens a;:d a1lo1rs the air ti:; be sucked into the cylinder and closes 2is the piston readies bottom dead cen(re. When the piston moves qpwJ.rds the z,i:- is
·2. Schem1tical diagram of a reciprocating compressor.
compressed and at a rated pressure the discharge valve opens to discharge the compressed air to the system. A double acting com- pressor uses both ends of the cylinder for suction and discharge, thus discharging approximately twice as much air per cylinder as the
single.acting unit. ···
Fig. 4,3 shows P-v and '1'-s diagrnm 01 thelheoreticai com- RPes~ion1 cycle neglecting the clearance volum~. Process P-1 represen'rs suction, process 1-2 the compression fr/om pressure pl to
"P