REVERSIBLE NON-FLOW PROCESS CONSTANT VOLUME PROCESS (ISOCHORIC PROCESS) In a constant volume process, he working substance is contained in a rigid

Transcription

1 REVERSIBLE NON-FLOW PROCESS CONSTANT VOLUME PROCESS (ISOCHORIC PROCESS) I a ostat olume roess, he workig substae is otaied i a rigid essel, hee the boudaries of the system are immoable, so work aot be doe o or by the system other tha arallel whih work iut whih is egatie ad this a be arried out i ratie by some method of durig the fluid. It will be assumed that ostat olume idies zero work uless otherwise stated. From the o-flow eergy equatio for uit mass, sie o work is doe, i.e. W = 0, we therefore hae Q = u -u Or for mass m, of the workig substae Q= U U That is, all the heat sulied i a ostat olume roess goes to ireasig the iteral eergy. This equatio a be writte i differetial form as Q = du For a erfet gas, Q = m (T -T ) The figures below show the - diagrams of ostat olume roess for a aour ad a erfet gas

2 CONSTANT PRESSURE PROCESS (ISOBARIC PROCESS) A losed system udergoig a roess at ostat ressure is illustrated i the figure below The fluid is elosed i a ylider by a isto o whih rests a ostat, weight. If heat is sulied, the fluid exads ad work is doe by the system i oeromig the ostat fore. Heat is extrated, the fluid otrats ad work is doe o the system by the ostat fore. From the o-flow eergy equatio, Q = (u -u ) + W Work might also be doe i the system simultaeously by turig the fluid with a addle, ad this egatie quatity of work must be iluded i the term W if o addle work is doe o the system ad the roess is reersible i.e. W = d = d, we hae Q = du + d Or Q d = du Sie is ostat, this a be itegrated to gie Q u u A further imliatio of the eergy equatio is ossible if a ew roerty is itrodued. Sie is ostat, d is idetial with d() Thus the eergy equatio a be writte as Q d Q d u du But seifi ethaly h = u+ i.e. Q = dh or i the itegrated form Q = h -h or for mass m, of a fluid

3 Q = H -H (reersible roess oly) That is the heat added i a reersible ostat ressure roess is equal to the irease i ethaly. The aboe equatio does ot aly if the roess is irreersible. For a erfet gas, Q = m (T -T ) The figures aboe show the - diagrams of ostat ressure roess for a aour ad a erfet gas NB: The areas shaded rereset the work doe by the fluid i.e. ( - ) CONSTANT TEMPERATURE PROCESS (ISOTHERMAL PROCESS) Whe the quatities of heat ad work are so roortioed durig a exasio or omressio that the temerature of the fluid remais ostat the roess is said to be isothermal whe a fluid i a ylider behid a isto exads from a high ressure to a low ressure, there is tedey for the temerature to fall. I isothermal exasio heat must be added otiuously i order to kee the temerature at the iitial alue. Similarly i a isothermal omressio heat must be remoed from the fluid otiuously durig the roess. It is ossible to show that for a reersible isothermal roess, a ertai defiite relatioshi must exist betwee ad, ad osequetly, the work doe has a defiite alue that a be redited. For ay reersible roess the eergy equatio is Q d = du Or Q - d u u The figures below show the - diagrams of isothermal roess for a aour ad a erfet gas

4 For isothermal roess of a erfet gas whih is assumed to hae a equatio of state. = RT=ostat, sie T = ostat i.e = ostat or = The work doe by a erfet gas whih udergoes a reersible isothermal exasio from state to state as show i the figure aboe is gie by the shaded area ad a be ealuated as: W = d I this ase = or = / W = = = [ ] = l( ) Sie = W = ( ) ( ) er uit mass of gas For mass, m, of the gas W = ( ) ( ) From, we hae W = ( ) ( ) Also Hee W = ( )

5 Ad for mass, m, of the gas W = mrt ( ) From Joule s law for erfet gas, du = dt For isothermal roess dt = 0 i.e du = 0 That is the iteral eergy remais ostat i a isothermal roess for a erfet gas From o-flow eergy equatio, we hae Q = du + W Sie du = 0 Q = W or Q = W That is, the quatity of heat sulied is equal to the work doe i a isothermal roess for a erfet gas oly. ADIABATIC PROCESS A adiabati roess is the oe i whih heat is reeted from rossig the boudary of the system. i.e. o heat is trasferred to or from the fluid durig the roess. That is, a adiabati roess is oe udergoe by a system whih is thermally isulated from its surroudigs. From the o-flow eergy equatio Q = u -u + W For a adiabati roess Q = 0, we hae -W = (u -u ) or W= (u -u ) This equatio is true for a adiabati o-flow roess whether or ot the roess is reersible. I a adiabati omressio roess all the work doe o the fluid goes to ireasig the iteral eergy of the fluid. Similarly i a adiabati exasio roess, the work doe by the fluid is at the exese of a redutio i the iteral eergy of the fluid.

6 A reersible adiabati roess is alled isetroi (ostat etroy) roess. For a aour udergoig a reersible adiabati roess the work doe a be foud from equatio gie aboe by ealuatig u ad u from tables. For a erfet gas, a law relatig ad has bee obtaied for a reersible adiabati roess as = ostat Eah erfet gas udergoig a reersible adiabati roess hae its ow alue of POLYTROPIC PROCESS The ostat olume ad ostat ressure roesses a be regarded as limitig ases of a more geeral tye of roess i whih both the olume ad ressure hage, but i a ertai seified maer. I may real roesses it is foud that the states durig a exasio or omressio a be desribed aroximately by a relatio of the form = ostat where is a ostat alled the idex of exasio or omressio, ad ad are aerage alues of ressure ad seifi olume for the system. Comressios ad exasio of the form = ostat are alled olytroi roesses. Whe = 0, the relatio redues to = ostat (isobari roess) ad whe =, it a be see to redue to = ostat by writig it i the form = ostat For ay reersible roess, W = d For a roess i whih = ostat, we hae Where is a ostat Thus, d d W ) ( Sie ostat = =

7 W This equatio is true for ay workig substae udergoig a reersible olytroi roess. It follows also that for ay olytroi roess, we a write The itegrated form of the eergy equatio for a reersible olytroi roess a therefore be writte as Q u u I a olytroi roess the idex deeds o i the heat ad work quatities durig the roess. The arious roesses osidered reiously are seial ases of the olytroi roess. Whe =, = or = ostat (isobari roess) Whe =, = or / = C i.e. = ostat (isohori) Whe =, = i.e. T= ostat (isothermal roess Sie /T=ostat for a erfet gas Whe =, = ostat i.e. reersible adiabati roess This is illustrated o a - diagram show below State to A is ostat ressure oolig ( = 0) State to B is isothermal omressio ( = ) State to C is reersible adiabati omressio ( = ) State to D is ostat olume heatig (= )

8 Similarly, State to A is isobari heatig State to B is isothermal exasio State to C is reersible adiabati exasio State to D is isohori oolig