Thermodynamics and Heat Engine4 (3+1)

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

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 x₁ and the final condition in the superheat region as point 2 at pressure p₂ (= p₁) and super heating temperature t_sup,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

 w_1-2 =                                            (kJ/kg)                

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

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

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

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

   δq = du + p.dv

          q_1-2 = (u_sup,2 – u₁) + ∫pdv = (u_sup,2 – u₁) + (p₂v_sup,2 – p₁v₁) = (h_sup,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.h_g,1 + (1 -x).h_f,1 = h_f,1 + x₁. h_fg,1    

                               where, h_g,1  and h_f,1 are the specific enthalpy  of saturated vapor and saturated liquid  respectively

                                            h_fg,1 = (h_g,1 – h_f,1) is latent heat of vaporization taken from steam table corresponding to pressure p₁ (p) _.

                  h_sup,2 is enthalpy of superheated steam at point ‘2’ corresponding to pressure p₂ (p) and temperature t_sup,2  taken from superheated steam table.

(d) Change in Entropy: 

            Δs = s_2sup,2 - s₁      

                   s₁ is specific entropy of wet steam at point ‘1’ =  x₁.s_g,1 + (1 –x₁).s_f,1 = s_f,2 + x₁. s_fg,1    

where.s_g,1  and s_f,1 are the specific entropy  of saturated vapor and saturated liquid  respectively.

                                          s_fg,1 = (s_g,1 – s_f,1) is specific entropy of vaporization taken from steam table corresponding to pressure p₁ (p)_.

    s_sup,2  is the entropy of superheated steam at point ‘2’ corresponding to pressure p₂ (p) and temperature t_sup,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 t_sup,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.v_g,2 + (1 –x₂).v_f,2                                                                                                                   ………………(22.1)

                where.v_g,2 and v_f,2   are the volume of saturated vapor and saturated liquid taken from steam table corresponding to pressure p_{2 .}

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

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

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

    From equations (22.1) and (22.2)

          v_sup,1 =   v₂ = x₂.v_g,2 + (1 –x₂).v_f,2   =  v_f,2  + x₂. v_g,2 – x₂. v_f,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,

              w_1-2  =  = 0.

(c) Heat Transfer: From First Law,

          δq = du + pdv       

or      q_1-2 = (u₂ – u_sup,₁) + ∫pdv

Since dv = 0         q_1-2 = (u₂ – u_sup,1)

or         q_1-2 = (h₂ – p₂v₂) + (h_sup,1- p_1.v_sup,1)                                                 ( kJ/kg)

                  h₂ is specific enthalpy of wet steam at point ‘2’ = x₂.h_g,2 + (1 –x₂).h_f,2 = h_f,2 + x₂. h_fg,2    

                             where h_g,2  and h_f,2 are the specific enthalpy  of saturated vapor and saturated liquid respectively

                                         h_fg,2 = (h_g,2 – h_f,2) is latent heat of vaporization taken from steam table corresponding to pressure p_2.

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

 (d) Change in Entrop: 

             Δs = s₂ – s_sup,₁      

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

s₂ is specific entropy of wet steam at point ‘2’  = x₂.s_g,2 + (1 –x₂).s_f,2 = s_f,2 + x₂. s_fg,2    

where s_g,2  and s_f,2 are the specific entropy  of saturated vapor and saturated liquid respectively corresponding to pressure p₂ taken from steam table_.

           s_fg,2 = (s_g,2 – s_f,2) is specific entropy of vaporization taken from steam table corresponding to pressure p_2.

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 t_sup,1 to the final condition of wet  steam at pressure p₂ and dryness fraction x_2S (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  w_1-2S =  (u_sup,1 – u_2s)                  for reversible process (isentropic)

                 w_1-2 =  (u_sup,₁ – u₂)                    for irreversible process

                      u_sup,1 = h_sup,1.- p_{1 .} v_sup,1

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

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

                      u₂ = h₂.- p₂v₂

                      u_2s = h_2s.- p₂v_2s

                              where,  h₂ = x₂.h_g,2 + (1 –x₂).h_f,2  =  h_f,2 + x₂. h_fg,2    

                                                  h_2s = x_2s.h_g,2 + (1 –x_2s).h_f,2   = h_f,2 + x_2s. h_fg,2           

                                       and

     v₂ = x₂.v_g,2 + (1 –x₂).v_f,2

     v_2s = x_2s.v_g,2 + (1 –x_2s).v_f,2

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

             h_fg,2 = (h_g,2 – h_f,2) is latent heat of vaporization taken from steam table corresponding to pressure p_2.

where, v_g,2  and v_f,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

                 w_lost = (w_1-2s – w_1-2 )

                         = (u_sup,1 – u_2s) –  (u_sup,₁ – u₂) = (u₂ – u_2s)

(c) Heat Transfer:

                          δq = 0

 (d) Change in Entropy: 

                            Δs = (s_2s – s_sup,₁) = 0   for reversible process (isentropic)

                            Δs = (s₂  – s_sup,₁)          for irreversible process

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

s₂ is specific entropy of wet steam at point ‘2’  = x₂.s_g,2 + (1 –x₂).s_f,2 = s_f,2 + x₂. s_fg,2    

s_2s is specific entropy of wet steam at point ‘2s’  = x₂.s_g,2 + (1 –x_2s).s_f,2 = s_f,2 + x_2s. s_fg,2    

 where, s_g,2  and s_f,2 are the specific entropy  of saturated vapor and saturated liquid respectively

                    s_fg,2 = (s_g,2 – s_f,2) is specific entropy of vaporization taken from steam table corresponding to pressure p_2.

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 t_sup,2  as shown in Fig. 22.4.

 

Fig. 22.4. Isothermal process

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

           δq = T.ds

          q_1-2 =                                 (kJ/kg) 

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

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

where s_g,1 and s_f,1   is the entropy of saturated vapor and saturated liquid taken from steam table corresponding to pressure p_{2 .}

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

w_1-2 = q_1-2 - (u_sup,₂– u₁)  =                   

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

                                 =  h_sup,2 − p₂v_sup,2

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

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

 u₁ is internal energy of wet steam at point ‘1’  = h₁ − p₁v₁

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

                                                                                                                            =  h_f,1 + x₁. h_fg,1      

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

       h_fg,1  is latent heat of vaporization taken from steam table corresponding to pressure p₁ = (h_g,1 – h_f,1).

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

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

(d) Change in Entropy:          

                      Δs = (s_sup,2  – s₁)         

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

s₁ is specific entropy of wet steam at point ‘1’  = x₁.s_g,1 + (1 – x₁).s_f,1  = s_f,2 + x₁. s_fg,1    

where, s_g,1  and s_f,1 are the specific entropy  of saturated vapor and saturated liquid respectively

            s_fg,1 = (s_g,1 – s_f,1) is specific entropy of vaporization taken from steam table corresponding to pressure p₁ (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)

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

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

where.v_g,2 and v_f,2   is the specific volume of saturated vapor and saturated liquid taken from steam table corresponding to pressure p_{2 .}

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

  δq = du + δw  =  du + pdv

  q_1-2 = (u₂ − u_sup,1) +                                       (kJ/kg)

           Here, u_sup,1 is the internal energy of superheated steam at point ‘1’ corresponding to pressure p₁ and temperature t_sup,1  = h_sup,1.- p₂v_sup,1

             and  u₂ = h₂ − p₁v₂

Therefore, q_1-2 = [(h₂  p_{1 .} v₂) − (h_sup,1  p_{2 .}v_sup,1)] +                 (kJ/kg)

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

                                                                                                                     =  h_f,2 + x₂. h_fg,2      

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

          h_fg,2  is latent heat of vaporization taken from steam table corresponding to pressure p₂ = (h_g,2 – h_f,2).

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

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

 (d) Change in Entropy:    

                          Δs = (s₂  – s_sup,₁)         

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

                                      s₂ is entropy of wet steam at point ‘2’  = x₂.s_g,2 + (1 – x₂).s_f,2 = s_f,2 + x₂. s_fg,2    

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

            s_fg,2 = (s_g,2 – s_f,2) is specific entropy of vaporization taken from steam table corresponding to pressure p_2. 

 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 (t_sup,₂). The final state 2 is set by p₂ and t_sup,₂ (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 t_sup,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 =  h_sup,2

or   h_f,1 + x₁. h_fg,1   =  h_sup,2

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

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

 h_fg,1 = is latent heat of vaporization taken from steam table corresponding to pressure p₁ = (h_g,1 – h_f,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 = (s_sup,2  – s₁)         

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

                                    s₁ = x₁.s_g,1 + (1 – x₁).s_f,1 = s_f,2 + x₁. s_fg,1    

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

            s_fg,1 = (s_g,1 – s_f,1) is sp. entropy of vaporization taken from steam table corresponding to pressure p_1.