Transient Calculations

Often, one needs to estimate the time taken to heat up a boiler knowing the flue gas flow and temperature or the tube wall temperature attained by a superheater without any flow of steam inside the tubes (say, during startup of the boiler). Lumped mass system analysis is used, which gives good estimates sufficient for practical purposes.

Gas at a temperature T g1 enters an evaporator that is initially filled with water at a temperature of t 1 . The following energy balance equation can be written neglecting the heat losses:

Md c t/dz = WC g pg ( T g 1 − T g 2 ) = UA T ∆ (6.38)

where M c is the water equivalent of the boiler = mass of steel × specific heat of steel +mass of water × specific heat of water (weight of boiler tubes, drums, and casing is included) dt/dz is the rate of change of temperature °C/h W g is the gas flow, kg/h

C pg is the gas specific heat, kcal/kg°C T g1 ,T g2 are gas temperatures entering and leaving the unit, °C.

U is the overall heat transfer coefficient, kcal/m 2 h °C

A is the surface area, m 2

∆T log mean temperature difference C = ,° =  ( T g 1 −− t )( T g 2 − t /ln T )   ( g g 1 − t/T )( g 2 − t ) 

(6.39) t is the temperature of water boiler, °C

Combining with the earlier equation, we have

 UA  =

  ( T g 2 − t )   WC g pg or

g 2 =+ ( g 1 − UA WgCpg / ) /e =+ t ( T g 1 − t /K

where Ke = UA WgCpg / (6.41) Substituting (6.40) in (6.38) and simplifying, we have

M dt/dz c = WC g pg T g 1 − tK − /K

Miscellaneous Boiler Calculations 347

dt/ T ( g 1 − t ) =  WC g pg ( K − 1 ) /M K dz c

Integrating from time t 1 to t 2 , we have

ln[[(T − t/T ) ( − t )] = [ WC ( K − 1 )( /MKz ) ] (6.44)

g 1 1 g 1 2 g pg

This equation gives an idea of the time required to heat up the boiler from temperature t 1 to, say, the boiling temperature of 100°C. Once steam starts forming additional equations for latent heat, flow-through vent may be developed to obtain a more accurate startup curve.

Example 6.19

A waste heat boiler weighs 50,000 kg and contains 30,000 kg of water at 27°C. To the boiler enters 130,000 kg/h of flue gases at 800°C. Assuming that gas specific heat =

0.3 kcal/kg°C, steel specific heat = 0.12 kcal/kg°C, surface area for heat transfer = 2000 m 2 , and overall adjusted U = 40 kcal/m 2 h °C, determine the time taken to heat the boiler with the water to 100°C.

Solution :

UA/W C g pg = × 40 2 000 130 000 0 3 , / ( , × . ) = .. 2 05 Ke = . 2 05 = . 7 78 M c = , 50 000 ×× . 0 12 30 000 1 36 000 + , ×= , ln  ( 800 27 − )( / 800 100 − )  = . 0 099 =  130 0 , 000 0 3 × . × ( . 7 78 1 36 000 7 78 − ) /, / .  z

= . 0 944 z or z = 01 . h or 6 min

One has to then check the time for generating steam and ensure if the superheater gas inlet temperature is reasonable. In practice, some differences may be seen due to the assumptions made such as a uniform U for the complete evaporator; the drums may have a lower value of U compared to the tubes.

Example 6.20

The time required to heat up a superheater in a gas turbine exhaust plant from 27°C to 480°C has to be determined. At 550°C, 68,000 kg/h of turbine exhaust gases enter the

HRSG. Assume the gas-side heat transfer coefficient is 60 kcal/m 2 h °C, gas specific heat = 0.286 kcal/kg°C, weight of the superheater is 2000 kg, and surface area = 180 m 2 .

Solution

M c = 2000 × 0.12 = 240 kcal/°C

Using Equations 6.41 and 6.44,

K exp =  60 × 180 / ( , 68 000 0 286 × . )  = e xp ( . 0 555 ) = . 1 74 ln  ( 550 − 27 ) / ( 5 5 50 − 480 )  = 68 000 0 286 × . × ( . 1 74 1 − ) . 1 74 240 orz = . 0 058 or 3 . 66 m i n

This gives an idea of how fast the metal gets heated up. The steam generation based on the calculations done as in earlier example may take much longer than 3.6 min. If the startup is frequent, one can start up the gas turbine on low load when the exhaust gas temperature will be much lower. This is the reason that with high gas inlet temperatures,

348 Steam Generators and Waste Heat Boilers: For Process and Plant Engineers

the superheater is shielded from the hot gases by a screen section that reduces the gas temperature entering it. Else, it can get oxidized without flow of steam. In fired boilers, the firing rate is controlled to keep the gas temperature lower at the superheater inlet if it is

a radiant superheater. Here again, a convective superheater with a screen section helps.

Example 6.21

With 450°C exhaust gas temperature and the same gas flow (no bypass stack), what temperature will the tube metal reach in 10 min?

ln[(450 − 27)/(450 − t)] = 68,000 × 0.286 × 0.74 × 0.167/(1.74 × 240) = 5.75 or t = 449°C. The superheater will reach the gas temperature in about 10 min. In 5 min, the tubes will attain 425°C. It is likely that steam will start flowing through the tubes within 10 min. As discussed in Appendix E, the finned evaporator weighs less and is compact, and hence, the water equivalent is lower compared to plain tube boilers, and hence, this helps to speed up the steam generation.

Transient Calculations