METHODS OF IMPROVING EFFICIENCY

The efficiency of the Rankine cycle can be improved by the following methods:

1. Reheating
2. Regeneration
3. Combination of reheating and regeneration

1 Reheat Cycle

Figure 4.8(a) shows the schematic diagram for the reheat cycle. The corresponding h–s and T–s diagrams are shown in Fig. 4.8(b) and Fig. 4.8(c), respectively. The steam is extracted at a suitable point and is reheated with the help of flue gases in the boiler furnace. This increases the dryness fraction of steam passing through the LP turbine.

Heat supplied, q_s = (h₁ − h_f6) + (h₃ − h₂)

Pump work, w_p = h_f 6 − h_f5

Turbine work, w_t = (h₁ − h₂) + (h₃ − h₄)

Net work output, w_net = w_t − w_p

Figure 4.8 Reheat cycle: (a) Schematic diagram, (b) h-s diagram, (c) T-s diagram

Figure 4.9 Reheat cycle with pressure drop between stages: (a) T-s diagram, (b) h-s diagram

If pump work is neglected,

2 Effect of Pressure Drop in the Reheater

The T-s and h–s diagrams for a reheat cycle with pressure drop is shown in Fig. 4.9(a) and (b), respectively. Steam enters the prime mover first stage at point 1. Line 1-2 is the isentropic process. However, in actual practice, it is an irreversible process due to internal frictional resistance offered to the flow of steam. Thus, the condition after the irreversible adiabatic expansion is shown by point 2′. The steam at condition 2′ is passed on the reheater. Since steam cannot pass through the reheater without a pressure drop, the exit from the reheater at point 3 account for the reheater pressure drop from p₂ to p₂ − Δp. The steam then expands irreversibly and adiabatically to point 4′.

For the first stage, turbine efficiency, 

And for the second stage, 

Example 4.4

A high pressure boiler delivers steam at 90 bar and 480°C. The steam is expanded in the first stage of turbine to 12 bar and withdrawn and passed on the reheater. The expansion now takes place in the second stage of the steam turbine, down to the condenser pressure of 0.07 bar. Calculate (a) the efficiency of the ideal cycle and (b) the work output and efficiency of the reheat cycle operating through the same states neglecting pump work.

Figure 4.10 Reheat Rankine cycle

Solution

The cycle on T-s diagram is shown in Fig. 4.10.

At 90 bar, 480°C, from steam tables,

h₁ = 3336.5 kJ/kg, s₁ = 6.5942 kJ/kg.K

Now, s₁ = s₂

Since, s_g at 12 bar is less than 6.5942, the state 2 is in the superheated region. then, we get

h₂ = 2814.4 kJ/kg, t₂ = 201°C

At 12 bar, 480°C, h₃ = 3432.8 kJ/kg, s₃ = 7.6198 kJ/kg.K

At 0.07 bar, s_f4 = 0.5589 kJ/kg.K, s_fg4 = 7.7179 kJ/kg.K

s₃ = s_f4 + x_4sfg4

or 7.6198 = 0.5589 + x₄ × 7.7179

or x₄ = 0.915

h₄ = h_f4 + x₄h_fg4 = 163.40 + 0.915 × 2409.1 = 2367.7 kJ/kg

h_fs = h_f4 = 163.4 kJ/kg

h₆ = h₅ + w_p

w_p = v_fs (p₁ − p₂) × 10²

= 0.001007 (90 − 0.07) × 10² = 9 kJ/kg

h₆ = 163.4 + 9 = 172.4 kJ/kg

1. q_a = (h₁ − h₆) + (h₃ − h₂) = (3336.5 − 172.4) + (3432.8 − 2814.4) = 3782.5 kJ/kgw_net = (h₁ − h₂) + (h₃ − h₄) − w_p= (3336.5 − 2814.4) + (3432.8 − 2367.7) − 9 = 1578.2 kJ/kg
2. Neglecting pump workq_a = 3782.5 + 9 = 3781.5 kJ/kgw_net = 1578.2 + 9 = 1587.2 kJ/kg

Example 4.5

In Example 4.4, steam enters the reheater at 12 bar, 210°C, and leaves at 11.5 bar, 480°C. The combined steam rate is 3 kg/kWh, the generator efficiency is 94%, and the heat loss through the turbine casing is 1% of the throttle enthalpy. Determine the turbine thermal efficiency and condition of exhausted steam.

Solution

The processes in the T-s diagram is shown in Fig. 4.11.

Figure 4.11 Reheat Rankine cycle with turbine efficiency

Refer to Fig. 4.11

 = 210°C,  = 12 bar, t₃ = 480°C, p₃ = 11.5 bar

Now h₁ = 3336.5 kJ/kg, h₂ = 2814.4 kJ/kg

 at 12 bar, 210°C from steam tables

= 2839.8 kJ/kg

h₃ at 11.5 bar, 480°C = 3433.4 kJ/kg

Now 

For energy balance of turbine,

h₁ + h₃ = h′₂ +  + w_net + q_loss

 = h₁ −  + h₃ − w_net − 0.01 × h₁

= 3336.5 − 2839.8 + 3433.4 − 1276.6 − 0.01 × 3336.5 = 2620.1 kJ/kg

At p₄ = 0.07 bar,  = 2620.1 kJ/kg,  = 64°C

Since saturation temperature at 0.07 bar is 39.02°C, the exhaust steam is superheated and the degree of superheat is 24.98°C.

Efficiency of turbine upto extraction point,

Overall efficiency of turbine,

Example 4.6

Steam at 100 bar and 500°C enters a prime mover that has one stage of reheat. The steam is exhausted from the prime mover at 0.07 bar and 85% dry. The net work developed by the prime mover is 1600 kJ/kg of steam. Calculate the thermal efficiency of the prime mover.

Solution

The cycle on T-s diagram is shown in Fig. 4.12.

Figure 4.12 Reheat Rankine cycle

Refer to Fig. 4.12.

From steam tables for p₁ = 100 bar, 500°C,

h₁ = 3373.6 kJ/kg

At 0.07 bar, h_f5 = 163.38 kJ/kg

h₄ = h_f4 + x₄ h_fg4

= 163.38 + 0.85 × 2409.2 = 2211.2 kJ/kg

w_net = 1600 kJ/kg

w_net = (h₁ − h₂) + (h₃ − h₄)

or h₃ − h₂ = w_net + h₄ − h₁ = 1600 + 2211.2 − 3373.6 = 435.6 kJ/hg

Thermal efficiency η_th =