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Nozzles and Diffusers
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Adiabatic Process
An adiabatic process is one in which no heat is transferred to or from the fluid during the processSuch a process can be reversible or irreversibleFor an adiabatic process to take place, perfect thermal insulation for the system must be available.Most of the turbo-machines operate adiabatically.
the work done W by the fluid is at the expense of a reduction in the internal energy of the fluid. Similarly in an adiabatic compression process all the work done on the fluid goes to increase the internal energy of the fluidIsentropic processReversible adiabatic process is known as isentropic processThere is no change in the entropy
Work done in isentropic processNon flow Process
Flow process
Specific workThe work capacity of a turbomachine per unit mass rate of flow of fluid is called the specific work (W) of the turbomachineIt is defined as the difference in the useful energy content per unit mass rate of flow medium between the two ends namely pressure and suction ends of the machineThe useful energy content of the flow medium can be expressed as the sum of pressure, kinetic and gravitational potential energiesWhen the change in density is negligible,
Units of W are m2/s2 or J/kgSpecific work is used as a common term for all types of turbo machines instead of Head or pressure ratioHead is commonly used for Pumps and hydraulic turbinesPressure ratio is commonly used for compressors and steam/gas turbinesThe above equation cannot be used for compressible flow as the density is not constant .Since by definition, W is the useful energy, it is the work of a the isentropic process
Energy equationFrom the first law of thermodynamics we have
For applications in turbomachines, the energy term will include internal energy, gravitational potential energy and kinetic energy.
Change in the energy in a finite process between two states is given by
Sub. In eqn. (1)
Dividing throughout by mass m,
Steady flow energy equation.For study flow work through turbomachines, the work term includes the shaft work and flow workThe SFEE can be written as
Since H = U + pV
In terms of specific quantities
Eqn. (4) & (5) is the SFEE for control volume or open systemThis can be rewritten for various processes in various turbomachines and their componentsHydro TurbomachinesHere =1/v = Constant , u1 u2 and q 0So we have from eqn (5)
Guide blades and draft tubesThese are stationary, so shaft work is zero, then eqn. (6) becomes
Compressible flow machinesMost of the compressible flow machines such as gas turbines, compressors and blowers are adiabatic machines, ie. q = 0In these machines change in PE also negligible as compared to others, so the eqn .(5) becomes
The shaft work is given by
If the entry and exit velocities are small or the difference between them is negligible, then the shaft work is given by the difference between the static enthalpies
Stagnation state & Stagnation propertiesA state defined by the stagnation temperature and pressure is the stagnation stateThe stagnation state of a gas is often used as a reference state.Stagnation enthalpyIt is defined as the enthalpy of the gas or vapour when it is adiabatically decelerated to zero velocity
The stagnation enthalpy is constant in a flow process that does not involve a work transfer or a heat transfer even though irreversible processes may be present
The stagnation state is represented by the point 01 brought about by an irreversible deceleration.For a reversible deceleration the stagnation point would be at point 01s and the state change would be called isentropic.
Stagnation temperatureThe stagnation temperature is defined through stagnation enthalpyFor a perfect gas we have
Where M is the Mach number The mach no. is defined as the ratio of local velocity of the gas to the local velocity of sound
Stagnation velocity of soundSince
Stagnation pressureIt is the pressure of gas or fluid is obtained by decelerating it in a reversible adiabatic(isentropic) process to zero velocityThe ratio of stagnation and static pressure can be obtained as follows
When the pressure changes are small, the process can be assumed to be incompressible. Then stagnation pressure can be determined from the Bernoulli eqn.
Adiabatic flow through nozzles
But the actual process is irreversible adiabatic. Since the shaft work is absent the stagnation enthalpy remains constant (h01=h02).
The actual velocity c2 is less tan than the isentropic velocity.Also irreversible adiabatic flow experiences an increase in entropy s and decrease in the stagnation pressure p0= p01 p02
Nozzle efficiency for large pressure ratioThe function of nozzle is to transform the energy of the expanding gas in to KE. Nozzle efficiency should be a measure of he energy transformation
We have
Nozzle Velocity coefficientMost of the applications the gas or steam enters the nozzle from a large space. So the enthalpy at the beginning of the process is considered as h01 instead of h1
We have
Nozzle efficiency for small pressure ratioFrom eqn. (7) we have
(h2-h2s) is the enthalpy loss due to irreversible flowFor isentropic process we have
If the change in pressure is small, the flow can be considered as incompressible ( = constant).
Adiabatic flow through diffusersThe transformation of KE of gases in to a static pressure rise takes place in a diffuser
There is no change in entropy and loss in stagnation pressure during a reversible adiabatic pressure.
But actual process is irreversible adiabatic, it must be accompanied by a stagnation pressure loss and increase in entropy.Final state after irreversible adiabatic process is represented by point 2. This is fixed by assuming same change in the KE.
There will be a change in static pressure but the stagnation enthalpy remains the same
Diffuser efficiency for small pressure riseFor small rise in static pressure, the flow in the diffuser can be considered as incompressible ( constant)
Therefore the pressure rise is given by
For the actual state point 2
Therefore the pressure rise in actual process is given by
But
Thus we have
Diffuser efficiency
But we have
Diffuser efficiency for large pressure riseFor large pressure rise through diffuser, density change is appreciable and flow is compressible
We have the energy equation for the processes
Substituting this in to the efficiency equation we have
For incompressible flow this expression can be reduced to expressions derived for small pressure rise.For example
Therefore we have
This is same as the expression that derived for small pressure rise.
Work and efficiencies in turbine stagesEnergy equation in various forms and the concepts discussed for nozzles can employed for turbinesIn turbines the difference is on account of the presence of shaft work
Ideal and actual expansion process in turbine
In the isentropic process (1-2s), on account of work transfer there is a drop in stagnation enthalpy and the entropy remains same on account of isentropic processThe actual expansion process (irreversible adiabatic) 1-2, on account of irreversibility there is an increase in entropyStagnation pressures at exit cannot be compared with initial stagnation pressures because of the work transfer.The actual work at the turbine can be determined from the change in stagnation enthalpies at the entry and exit.
For perfect gas
Total to total EfficiencyEfficiency of turbine is ratio of actual to ideal work for the same Pressure ratio (pr=p1/p2)In turbines the actual work can be measured and the ideal work is hypothetical and depends on the manner it is definedIf the ideal work is defined as the work obtained during the isentropic expansion from the stagnation state O1 to O2s , then the efficiency based on this is known as the total to total efficiency
Here the KE of the gas at the exit is not considered as wasted since it is contained in the term h02s , H02s = h2s + c2s2/2
The stagnation pressure lines for p02s and p02 are different. However the distance between them is small.The stagnation pressure ratio is
This expression when substituted in eqn. 8
For given stagnation temperature, pressure ratio and efficiency the output power at the shaft is
Total to static efficiencySome turbines the KE of the out going jet is lost because it is not used after the turbineeg. Some turbine stages exhaust in to atmosphere or in a closed space like condenserIn such case the ideal work is the work done between the states O1 and 2s
The actual work remains the same as the beforeThe total to static efficiency is given by
If
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