Tuesday, October 31, 2023

LESSON - 22 HEATING AND EXPANSION OF VAPOUR IN NON-FLOW PROCESSES

 22.1  HEATING AND EXPANSION OF VAPOR IN NON-FLOW PROCESSES

For various thermodynamics processes, vapor cannot be treated in the same way as a gas because vapor does not follow laws relating to gases.  Therefore the analysis of various processes for vapor is required. However, the following basic energy equations derived from First and Second Laws for the vapor are the same as those deduced for a perfect gas.

δq = du + p.dv,        δq = dh – v.dp      and          δq = T.ds

The equations for work done  during non-flow process,  during flow process, steady flow energy equation, entropy balance, etc. are also be applied to vapor in the same way it has been applied to gases.

Once a vapor becomes superheated the various processes of heating and expansion of gases may be applied to vapor as vapor in superheated state will approximately follow the laws of gases except well below its critical temperature.

The various thermodynamic processes for vapor are given in the subsequent subheadings.


22.1.1. Constant Pressure Process:

(a)   p-v, T-s, and h-s Representation. Assume that steam undergoes a constant pressure heating process from its initial state to final state.  Let the initial condition of steam be in the wet region as point 1 at pressure p1 having dryness fraction xand the final condition in the superheat region as point 2 at pressure p(= p1) and super heating temperature tsup,2 as shown in Fig. 22.1 on p-v, T-s and h-s diagrams.

 Fig. 22.1. Constant pressure heating process of steam 

(b) Work done: The work done during a reversible non-flow process at constant pressure is given by

 w1-2 =                                            (kJ/kg)                

       v1  is specific volume of wet steam at point ‘1’  = (1 –x1).vf,1   + x1.vg,1

              where, vg,1 ,vf,1 are the specific volumes of saturated vapor and saturated liquid  respectively as taken from steam table corresponding to pressure p1 (or p2) .

       vsup,2 is the volume of superheated steam at point ‘2’ corresponding to pressure p and temperature tsup,2  taken from superheated steam table.

(c) Heat Transfer: The heat transfer during non-flow process is given by

   δq = du + p.dv

          q1-2 = (usup,2 – u1) + ∫pdv = (usup,2 – u1) + (p2vsup,2 – p1v1) = (hsup,2 – h1)           (kJ/kg)

          Thus the heat transfer at constant pressure is equal to change in enthalpy.

                   h1 is specific enthalpy of wet steam at point ‘1’ =  x.hg,1 + (1 -x).hf,1 = hf,1 + x1. hfg,1    

                               where, hg,1  and hf,1 are the specific enthalpy  of saturated vapor and saturated liquid  respectively

                                            hfg,1 = (hg,1 – hf,1) is latent heat of vaporization taken from steam table corresponding to pressure p1 (p) .

                  hsup,2 is enthalpy of superheated steam at point ‘2’ corresponding to pressure p(p) and temperature tsup,2  taken from superheated steam table.

(d) Change in Entropy: 

            Δs s2sup,2 - s1      

                   s1 is specific entropy of wet steam at point ‘1’ =  x1.sg,1 + (1 –x1).sf,1 = sf,2 + x1. sfg,1    

where.sg,1  and sf,1 are the specific entropy  of saturated vapor and saturated liquid  respectively.

                                          sfg,1 = (sg,1 – sf,1) is specific entropy of vaporization taken from steam table corresponding to pressure p(p).

    ssup,2  is the entropy of superheated steam at point ‘2’ corresponding to pressure p2 (p) and temperature tsup,2  taken from superheated steam table.

 For cooling process at constant pressure the nomenclature used for states in Fig. 22.1 will be interchanged.

22.1.2. Constant Volume Process:

(a)    p-v, T-s and h-s Representation: Assume that steam undergoes a constant volume cooling process from the initial condition of superheated steam and be defined by pressure p1 and super heating temperature tsup,1 to the final condition of steam at pressure p2 and dryness traction x2 as shown on p-v and T-s diagrams in Fig. 22.2.

Fig. 22.2. Constant volume cooling process of steam

        v2 = x2.vg,2 + (1 –x2).vf,2                                                                                                                   ………………(22.1)

                where.vg,2 and vf,2   are the volume of saturated vapor and saturated liquid taken from steam table corresponding to pressure p2 .

     But vsup,1 = v2                                                                                                                                     …   ………….(22.2)

               where, v2  is specific volume of wet steam at point ‘2’  

               where, vsup,1  is the volume of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  taken from superheated steam  table.

    From equations (22.1) and (22.2)

          vsup,1 =   v2 = x2.vg,2 + (1 –x2).vf,2   =  vf,2  + x2. vg,2 – x2. vf,2

    Therefore,    x2 = 

This gives the quality of steam at the final condition ‘2’ hence the point ‘2’ on T-s diagram can early be located.

(b)   Work done: As the volume remains constant, dv = 0. Thus the non-flow work,

              w1-2  =  = 0.

(c) Heat Transfer: From First Law,

          δq = du + pdv       

or      q1-2 = (u2 – usup,1) + ∫pdv

Since dv = 0         q1-2 = (u2 – usup,1)

or         q1-2 (h2 – p2v2) + (hsup,1- p1.vsup,1)                                                 ( kJ/kg)

                  h2 is specific enthalpy of wet steam at point ‘2’ = x2.hg,2 + (1 –x2).hf,2 = hf,2 + x2. hfg,2    

                             where hg,2  and hf,2 are the specific enthalpy  of saturated vapor and saturated liquid respectively

                                         hfg,2 = (hg,2 – hf,2) is latent heat of vaporization taken from steam table corresponding to pressure p2.

                 hsup,1  is the enthalpy of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

 (d) Change in Entrop: 

             Δs = s2 – ssup,1      

ssup,1  is the entropy of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

s2 is specific entropy of wet steam at point ‘2’  = x2.sg,2 + (1 –x2).sf,2 = sf,2 + x2. sfg,2    

where sg,2  and sf,2 are the specific entropy  of saturated vapor and saturated liquid respectively corresponding to pressure ptaken from steam table.

           sfg,2 = (sg,2 – sf,2) is specific entropy of vaporization taken from steam table corresponding to pressure p2.

For heating process at constant volume the nomenclature used for states in Fig. 22.2 will be interchanged.

22.1.3. Adiabatic Process (Reversible and Irreversible):

(a)   T–s and h-s Representation: Assume that steam undergoes an adiabatic expansion process from the initial condition of superheated steam and be defined by pressure p1 and super heating temperature tsup,1 to the final condition of wet  steam at pressure p2 and dryness fraction x2S (reversible adiabatic) or x2 (irreversible adiabatic).

On h-s and T-s diagrams shown in Fig. 22.3:

Process ‘1-2s’   Reversible adiabatic (i.e. isentropic)  Showing no increase in entropy.

Process ‘1-2’     Irreversible adiabatic  Showing increase in entropy.


Fig. 22.3. Adiabatic process (Expansion)

(b) Work done: The work done for non-flow reversible/irreversible adiabatic process is given by

        δw  = δq - du

Since the process is adiabatic, δq = 0

        δw  = 0 - du = du

therefore  w1-2S =  (usup,1 – u2s)                  for reversible process (isentropic)

                 w1-2 =  (usup,1 – u2)                    for irreversible process

                      usup,1 = hsup,1.- p1 . vsup,1

where, hsup,1 is the enthalpy of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

 vsup,1  is the specific volume of superheated steam at point ‘1’ corresponding to pressure p1 and temperature Tsup,1  taken from superheated steam table.

                      u= h2.- p2v2

                      u2s = h2s.- p2v2s

                              where,  h2 = x2.hg,2 + (1 –x2).hf,2  =  hf,2 + x2. hfg,2    

                                                  h2s = x2s.hg,2 + (1 –x2s).hf,2   = hf,2 + x2s. hfg,2           

                                       and

     v2 = x2.vg,2 + (1 –x2).vf,2

     v2s = x2s.vg,2 + (1 –x2s).vf,2

where, hg,2  and hf,2 are the specific enthalpy  of saturated vapor and saturated liquid respectively taken from steam table corresponding to pressure p2.

             hfg,2 = (hg,2 – hf,2) is latent heat of vaporization taken from steam table corresponding to pressure p2.

where, vg,2  and vf,2 are the specific enthalpy  of saturated vapor and saturated liquid respectively taken from steam table corresponding to pressure p2.

Hence , work lost due to irreversibility is given by

                 wlost = (w1-2s – w1-2 )

                         = (usup,1 – u2s) –  (usup,1 – u2) = (u– u2s)

(c) Heat Transfer:

                          δq = 0

 (d) Change in Entropy: 

                            Δs = (s2s – ssup,1) = 0   for reversible process (isentropic)

                            Δs = (s – ssup,1)          for irreversible process

ssup,1  is the entropy of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

s2 is specific entropy of wet steam at point ‘2’  = x2.sg,2 + (1 –x2).sf,2 = sf,2 + x2. sfg,2    

s2s is specific entropy of wet steam at point ‘2s’  = x2.sg,2 + (1 –x2s).sf,2 = sf,2 + x2s. sfg,2    

 where, sg,2  and sf,2 are the specific entropy  of saturated vapor and saturated liquid respectively

                    sfg,2 = (sg,2 – sf,2) is specific entropy of vaporization taken from steam table corresponding to pressure p2.

22.1.4. Isothermal Process:

(a) p-v, T-s and h-s Representation: Assume that steam undergoes an isothermal expansion process from the initial condition of wet steam at pressure p1 and dryness fraction x1 to the final condition of superheated steam and be defined by pressure p2 and super heating temperature tsup,2  as shown in Fig. 22.4.

 

Fig. 22.4. Isothermal process

 (b) Heat Transfer: The heat transfer is given by

           δq = T.ds

          q1-2 =                                 (kJ/kg) 

ssup,2  is the entropy of superheated steam at point ‘2’ corresponding to pressure p2 and temperature tsup,2  taken from superheated steam table.

s1 is specific entropy of wet steam at point ‘1’  = x1.sg,1 + (1 –x1).sf,1

where sg,1 and sf,1   is the entropy of saturated vapor and saturated liquid taken from steam table corresponding to pressure p2 .

Thus the area under T-s diagram shown by shaded area gives the heat transfer.

 

(c) Work done: For non-flow process, the work done is given by

δw  = δq - du

w1-2 = q1-2 - (usup,2– u1)  =                   

usup,2 is the internal energy of superheated steam at point ‘2’ corresponding to pressure p2 and temperature tsup,2 

                                 =  hsup,2 − p2vsup,2

 where hsup,2  is the enthalpy of superheated steam corresponding to pressure p2 and temperature tsup,2  taken from superheated steam table.

           vsup,2  is the specific volume of superheated steam corresponding to pressure p2 and temperature tsup,2  taken from superheated steam table.

 uis internal energy of wet steam at point ‘1’  = h− p1v1

                  where,  h1 is enthalpy of wet steam at point ‘1’  = x1.hg,1 + (1 –x1).hf,1 

                                                                                                                            =  hf,1 + x1. hfg,1      

  where, hg,1  and hf,1 are the specific. enthalpy  of saturated vapor and saturated liquid respectively taken from steam table corresponding to pressure p1

       hfg,1  is latent heat of vaporization taken from steam table corresponding to pressure p= (hg,1 – hf,1).

                                                              v1 is specific volume of wet steam at point ‘1’  = x1.vg,1 + (1 – x1).vf,1

                                                               where, vg,1 and  vf,1     are the specific volume of saturated vapor and saturated liquid taken from steam table corresponding to pressure p1

(d) Change in Entropy:          

                      Δs = (ssup,2  – s1)         

ssup,2  is the entropy of superheated steam corresponding to pressure p2 (p) and temperature tsup,2  taken from steam table.

s1 is specific entropy of wet steam at point ‘1’  = x1.sg,1 + (1 – x1).sf,1  = sf,2 + x1. sfg,1    

where, sg,1  and sf,1 are the specific entropy  of saturated vapor and saturated liquid respectively

            sfg,1 = (sg,1 – sf,1) is specific entropy of vaporization taken from steam table corresponding to pressure p(p).

22.1.5. Polytropic Process:

 

(a) p-v Representation: Assume that steam undergoes a polytropic expansion process following the law pvn = c from pressure p1 to p2. It is to be noted that steam does not behave as perfect gas obeying pv = RT. The equation pvn= c is merely a statement of pressure-volume relationship during a reversible polytropic process. The non-flow polytropic process has application in steam engines, etc. This process is represented as 1-2 on p-v diagram shown in Fig. 22.5 in which the initial condition is superheated steam at pressure p1 and super heating temperature tsup,1 to the final condition of wet  steam at pressure p2 and dryness traction x2.

 

 

Fig. 22.5. Polytropic process

 (b) Work done: For non-flow process neglecting changes in KE and PE, the work done is given by

                                                           (kJ/kg)

vsup,1  is the specific volume of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

v2 is specific volume of wet steam at point ‘2’  = x2.vg,2 + (1 – x2) vf,2

where.vg,2 and vf,2   is the specific volume of saturated vapor and saturated liquid taken from steam table corresponding to pressure p2 .

(c) Heat Transfer: The heat transfer is given by

  δq = du + δw  =  du + pdv

  q1-2 = (u− usup,1) +                                       (kJ/kg)

           Here, usup,1 is the internal energy of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  = hsup,1.- p2vsup,1

             and  u= h2 − p1v2

Therefore, q1-2 = [(h2  p1 . v2) − (hsup,1  p2 .vsup,1)] +                 (kJ/kg)

                                                  h2 is enthalpy of wet steam at point ‘2’  = x2.hg,2 + (1 – x2).hf,2 

                                                                                                                     =  hf,2 + x2. hfg,2      

where, hg,2  and hf,2 are the specific enthalpy  of saturated vapor and saturated liquid respectively taken from steam table corresponding to pressure p2

          hfg,2  is latent heat of vaporization taken from steam table corresponding to pressure p= (hg,2 – hf,2).

                                                  vsup,1  is the specific volume of superheated steam corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

                                                  hsup,1  is the enthalpy of superheated steam at point ‘1’ corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

 (d) Change in Entropy:    

                          Δs = (s – ssup,1)         

                                      ssup,1  is the entropy of superheated steam corresponding to pressure p1 and temperature tsup,1  taken from superheated steam table.

                                      s2 is entropy of wet steam at point ‘2’  = x2.sg,2 + (1 – x2).sf,2 = sf,2 + x2. sfg,2    

where, sg,2  and sf,2 are the specific entropy  of saturated vapor and saturated liquid respectively taken from steam table corresponding to pressure p2.

            sfg,2 = (sg,2 – sf,2) is specific entropy of vaporization taken from steam table corresponding to pressure p2. 

 22.1.6. Throttling Process:

As described earlier, the throttling process involves passing of a higher pressure fluid through a narrow constriction resulting in reduction in pressure and temperature, increase in specific volume, increase in entropy and without any change in enthalpy.

The important characteristic of the throttling process is that enthalpy remains constant.

The process is adiabatic and no heat flow from or to the system but it not reversible.

Examples:  Steam stop valve and throttle valve installed at the entry of steam turbine in power plants. When flow of steam takes place through these valves, throttling process occurs.

Fig. 22.6. Throttling of steam

Refer Fig. 22.6. Let us assume that the initial condition of the steam is wet represented as point 1 corresponding to pressure p1 and unknown dryness fraction, x1. This steam is allowed to throttle to pressure p2 resulting in the superheating of steam (tsup,2). The final state 2 is set by pand tsup,(known as it can be measured). Since enthalpy remains constant, the initial state 1 can be found on h-s chart by drawing a horizontal line from a known point ‘2’ (corresponding to pressure p2 and temperature tsup,2) until it intersects the constant pressure p1 line. The dryness fraction line passing through ‘1’ gives the quality of steam (x1). It is obvious from the h-s chart that entropy increases during throttling process.

For throttling process, we have

       h1 =  hsup,2

or   hf,1 + x1. hfg,1   =  hsup,2

where hsup,2  is the enthalpy of superheated steam corresponding to pressure p2 and temperature tsup,2  taken from superheated steam table.

.hg,1  and hf,1 are the specific enthalpy  of saturated vapor and saturated liquid  respectively taken from steam table corresponding to pressure p1.

 hfg,1 = is latent heat of vaporization taken from steam table corresponding to pressure p= (hg,1 – hf,1).

The value of dryness fraction x1 may be calculated from the above equation. The throttling process is, therefore, used in determining the dryness fraction of steam in power plants.

(a) Change in Entropy:        

                              Δs = (ssup,2  – s1)         

                                    ssup,2  is the entropy of superheated steam corresponding to pressure p2 and temperature tsup,2  taken from superheated steam table.

                                    s1 = x1.sg,1 + (1 – x1).sf,1 = sf,2 + x1. sfg,1    

where, sg,1  and sf,1 are the specific entropy  of saturated vapor and saturated liquid respectively taken from steam table corresponding to pressure p1.  

            sfg,1 = (sg,1 – sf,1) is sp. entropy of vaporization taken from steam table corresponding to pressure p1.

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