Empirical Formula for Furnace Duty Estimation

In European and Russian boiler calculation practice, the Gurvich method has found wide acceptance. The following equation relates the furnace exit gas temperature T e with com-

bustion temperature T c [2].

0 . 6 0 . 6 c (2.4) ( Bo

+ Ma f

where

T e ,T c are the furnace exit gas temperature and adiabatic combustion temperatures M is an empirical factor impacted by fuel and gas temperature profile in the furnace

B o is the Boltzman number

B = ϕ WV f o c (2.5)

( σψ ApT ) e

where

V c is the average heat capacity of the flue gas in the furnace in the range of (T c –T e ) formed by 1 kg of fuel burned, kJ/kg K φ is the furnace efficiency or (1-% heat loss/100) and is typically 0.995

σ is the Stefan–Boltzman constant = 5.67 × 10 −11 kW/(m 2 K 4 )

ψ is an empirical factor = xζ ζ is 0.55 for fuel oil and 0.65 for gas

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

For coal firing, it varies from 0.35 to 0.55. For refractory-covered surfaces, it is 0.1–0.2. ζ is corrected by a factor y dependent on the furnace wall tube spacing to diameter. For pitch to diameter S T /d of 1.25, it is 0.97, and for S T /d of 2, it is 0.82. If the furnace has different sections with different tube spacing or partly covered with refractory, one may use the following weighted average value of ψ.

ψ = ∑yi Api ζi/Ap (2.6)

a f = coefficient of thermal radiation = 1/[1 + {1/∈ f − 1}ψ] (2.7) Flame emissivity ∈ f has been discussed earlier for nonluminous and luminous flames.

(2.8) where A and B are empirical coefficients depending on fuel fired. For gas and oil, A = 0.54

M = a coefficient = A − BX

and B = 0.2. For coal, A = 0.59 and B = 0.5. X is the relative position of the highest temperature zone in the furnace (Figure 2.9a). If x 1 is the distance of burner 1 from bottom of

Burner 1 x1

Temperature (°F) 3250

(a) Temperature profiles along furnace length or height (x1, x2 measured from furnace bottom). (b) CFD model- ing of furnace temperature profile.

Steam Generator Furnace Design

furnace and x is the distance from bottom of furnace to furnace exit, then X is the relative position = x 1 /x. If the burner is in the front wall as in a D-type boiler, one may use X = 0 as x 1 is 0. Chapter 1 shows how one may estimate the adiabatic combustion temperature T c for various fuels from excess air.

⋅ [{. − M 5 67 1 0 11 ψ AaT/WV 3 ( ϕ )} 0 . 6

c pfc

T c is obtained through combustion calculations as shown in Chapter 1. It is a function of

excess air, combustion air temperature, and fuel analysis. W f = fuel input kg/s.

Example 2.3

Solve Example 2.1 using Gurvich method. The boiler generates 45,372 kg/h (100,000 lb/h)

steam at 25 kg/cm 2 g (355 psig) using 105°C (221°F) feed water. Natural gas at 15% excess

air is fired as in Example 2.1. Compute also the energy absorbed in the furnace. D-type boiler with burner at front walls is used. The furnace is completely water cooled. No refractory is used. Pitch-to-diameter ratio of tubes is 1.75. LHV = 49,884 kJ/kg. Ambient air enters the furnace, and hence, additional heat input by air is not considered.

Solution

Data such as flue gas and combustion temperature and ratio of flue gas to fuel are obtained from combustion calculations (Chapter 1). Assume boiler heat losses = .5%.

Hence, φ = 0.995. Furnace projected area based on using 253 kW/m 2 net heat input is 127 m 2 . Adiabatic combustion temperature = 1780°C = 2053 K. Fuel quantity = 2344 kg/h = 0.651 kg/s.

Flue gas quantity = 49,216 kg/h = 13.67 kg/s. Ratio of flue gas to fuel = 49,216/234 = 21.

ψ = 0.88 × 0.65 = 0.572. ∈ f = .308 from Example 2.1. a f = 1/[1 + {1/0.308 − 1}]0.572] = 0.437. Assume T e = 1150°C. From the properties of flue gases, enthalpy of gases (from Appendix F) at 1780°C and 1150°C = 565 kcal/kg and 342 kcal/kg. Hence, V c = 21 ×

(565 – 342)/(1780 – 1150) = 7.43 kcal/kg K = 31.1 kJ/kg K. M = 0.54 as the burners are on front wall. Hence, T e /T c = 1/[M{5.67 × 10 –11 ψ A p a f T c 3 /(φW f V c )} 0.6 + 1] = 1/[0.54{5.67 × 10 –11 × 0.572 × 127 × 0.437 × 2053 3 /(0.995 × 0.651 × 31.1)} 0.6 + 1] = 0.683 or T e = 2053 × 0.683 = 1402 K =

1129°C. Enthalpy of flue gas at 1129°C from Appendix F is 335.5 kcal/kg = 1404.7 kJ/kg.

Hence, furnace duty Q f = 0.651 × 49,915−21 × .651 × 1404.7 = 13,291 kW (45.36 MM Btu/h). The average heat flux is then Q f /A p = 13,291/127 = 104.6 kW/m 2 (33,055 Btu/ft 2 h). To

estimate the fin tip temperature or CHF-related issues, the maximum heat flux consid- ering the variation in temperature profile of the flue gas is considered.

One has to increase the furnace area in case the furnace exit gas temperature is high and is likely to cause slagging problems at the furnace exit. However, the example is shown to illustrate the methodology.

Example 2.4

Estimate the furnace exit gas temperature using the Gurvich method when fuel oil is fired in the same furnace. LHV of fuel oil = 43,059 kJ/kg. Fuel quantity = 0.755 kg/s. Use the data from Chapter 1. Fifteen percent excess air is used. Air enters at ambient temperature.

Adiabatic combustion temperature = 1866°C = 2139 K and ratio of flue gas to fuel =

17.83 from Chapter 1. Fuel quantity = 2718 kg/h = 0.755 kg/s. ψ = 0.88 × 0.55 = 0.484. ∈ f = 0.473 from Example 2.2. a f = 1/[1 + {1/0.473 − 1}]0.484] = 0.65

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

Assume T e = 1100°C. From the properties of flue gases, enthalpy of gases (from Appendix F) at 1866°C and 1100°C = 572 kcal/kg and 314.7 kcal/kg. Hence, V c = 17.83 ×

(572 – 314.7)/(1866 − 1100) = 5.99 Kcal/kg K = 25.07 kJ/kg K. M = 0.54 as the burners are on front wall.

Hence, T e /T c = 1/[M{5.67 × 10 –11 ψ A p a f T c 3 /(φW f V c )} 0.6 + 1] = 1/[0.54{5.67 × 10 –11 × 0.484 × 127 × 0.65 × 2139 3 /(0.995 × 0.755 × 25.07)} 0.6 + 1] = 0.6267 or T e = 2139 × 0.6267 = 1341 K = 1068°C. Enthalpy of gas at 1068°C is 304.4 kcal/kg = 1274.5 kJ/kg. Furnace duty Q f = 0.755 × 43,059–17.83 × 0.755 × 1274.5 = 15,352 kW (52.4 MM Btu/h).

The heat flux and furnace duty are higher on oil firing. Hence, the circulation is checked based on heat flux obtained in oil-fired case. The furnace exit gas temperature is also lower on oil firing compared to natural gas. When a boiler is fired with both fuel oil and natural gas, it is common practice to design the boiler and superheater surfaces on oil firing to ensure the desired steam temperature is obtained and, when firing natural gas, the superheater steam temperature will only be higher, which can be controlled by various methods as discussed in Chapter 3.

(Note that the terms furnace heat input and furnace duty are different. The duty is used to compute the heat flux inside the tubes and to check for departure from nucleate boiling [DNB]. This is explained later).

It may be noted that the furnace calculations give an idea of the exit gas temperature, and accuracy can vary depending on burner type and location. Boiler designers obtain field data on the performance and work backward the furnace exit gas temperature and arrive at correcting curves. Unlike convective heat exchanger calculations where the correlations for heat transfer are well established and more reliable, furnace performance evaluation is still a gray area. The error in estimation of the furnace exit gas temperature could be as high as ±100°C (180°F).

Empirical Formula for Furnace Duty Estimation