Circulation Calculations

Circulation calculation in a natural circulation boiler is an iterative process.

1. First, a CR is assumed based on experience and type of steam generator and pressure. For low-pressure boilers (<70 barg), CR could range from 12 to 35, while for higher-pressure units, it can range from 5 to 15. CR = 1/x where x is the steam quality, fraction. Flow through the downcomers, evaporators, and risers is CR x steam generated.

2. The feed water temperature entering the drum should be known from thermal calculations, which should have already been done. This includes the complete boiler performance as illustrated in Chapter 3, furnace calculations, and estimation of energy absorbed in each parallel path such as front wall, rear wall, and side

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

walls. The resistance of each path must also be known. Mixture enthalpy entering the downcomers is calculated through an energy balance around the drum, Figure 2.13a.

(2.20) where h fw ,h e ,h m , and h g are the enthalpy of feed water, steam–water mixture

h fw + CR × h e =h g + CR × h m

leaving the evaporator, water entering the downcomers, and saturated steam, respectively.

It should be noted that cool water entering the downcomer aids better circulation. If

h m is close to or at saturation, then bubbles will be formed in the region near the entry of downcomer pipes, and circulation will be hindered. There may be several downcomers in

a boiler each with a different length and pipe size. A flow and pressure balance calculation is done to arrive at the flow in each downcomer and riser. As the downcomer flow enters the evaporator, it gets heated, and boiling starts after a distance from the bottom of the furnace. This is called the boiling height, L b .

L b = L × CR × W s (h f −h m )/Q (2.21) (All in consistent units)

h f is the enthalpy of saturated liquid Q is the energy absorbed in the furnace L is the furnace height W s is the steam generated by the boiler

1. The static head available is first estimated. It is typically the distance from the drum water level to the bottom of the furnace; then the following losses are estimated:

2. Losses in the downcomers including entry and exit losses.

3. Losses in the boiling height and two-phase losses such as friction, acceleration,

and gravity using Thom’s method or similar well-known two-phase correlations.

4. If there are unheated risers, their losses are estimated. Again there may be several

parallel paths with different pipe sizes and lengths. Flow in each parallel path is evaluated by estimating the pressure drop in each path for different flows and ensuring that the pressure drop is the same. This is an iterative process. These calculations are preferably done using a computer program.

5. Loss in drum internals. This is impacted by the presence of cyclones inside the

drum for steam separation. The static head is balanced with various losses, and if they match, the CR assumed is fine;

else, another CR is assumed and the calculations are repeated. If there are multiple parallel paths (a D-type boiler has front wall, rear wall, side wall, and boiler bank tubes), then flow and pressure balance calculations are carried out to ensure that the total losses in each parallel path are the same. CR in each parallel path may differ depending on its energy absorbed, tube size used, and resistance to flow.

Once the circulation calculations are done, the flow in each evaporator tube will be known along with the steam quality at its exit. Checks for heat flux in each path and DNB

Steam Generator Furnace Design

based on exit quality are then done. The variation of heat flux along flame length and view factors based on tube spacing enter into these calculations. If there is a potential for DNB, then efforts are taken to revise the downcomer and riser system pipe sizes to improve the CR in the boiler; else, the heat flux in the furnace has to be reduced, and the boiler design is redone. Based on experience and field data, one should be able to arrive at a good system considering these concerns.

Example 2.9

A waste heat boiler similar to the system shown in Figure 2.13a has the following data: steam pressure = 55.1 barg (800 psig). Static head available (from drum centerline to bottom of evaporator bundle) = 8 m. Steam drum is located far away from the evaporator, which is generating 58 t/h of steam; feed water temperature is 120°C.

Two downcomers of ID 285 mm and length 14 m and with six 90° bends are feeding the evaporator with two risers of ID 317 mm and of length 11 m with five bends are feeding drum. The evaporator bundle is 6 m high and has 420-44 mm ID tubes. Determine

the approximate CR in the evaporator. Drum internals loss = 352 kg/m 2 .

Solution

First, the energy balance is performed in the drum assuming a CR. Say CR = 11. At 55.1 bar, from steam tables, v g = 0.035 and v f = 0.0013 m 3 /kg. Enthalpy of feed water h f = 121.2 kcal/kg, enthalpy of saturated steam h g = 666.1 kcal/kg, and saturated water h l = 284.27 kcal/kg. Saturated steam temperature = 271°C. Assume CR = 11. Mixture enthalpy leaving the drum h e = 0.0909 × 666.1 + 0.9091 × 284.27 = 319 kcal/kg. From energy balance in the drum, 11 × 319 + 121.2 = 666.1 + 11h m or h m = 269.5 kcal/kg. From steam tables, this corre- sponds to 259°C and v m = 0.001271 m 3 /kg. Hence, the water will start boiling after some distance from the bottom as the saturated steam temperature is 271°C.

1. Static head available = 8/.001271 = 6294 kg/m 2. 2. To compute the losses in downcomers, the velocity of water in downcomer must be known. V dc = 1.246wv m /d i 2 (see Appendix B for formula for velocity inside tubes). Flow in each downcomer w = 11 × 58,000/2/3,600 = 88.6 kg/s. Then V dc = 1.246 × 88.6 × 0.001271/0.285/0.285 = 1.73 m/s. Velocity head VH = V 2 /(2gv m ) = 1.73 × 1.73/2/9.8/0.001271 = 120 kg/m 2 (g = 9.8 m/s 2 ). The entrance

plus exit loss in downcomer = 1.5VH = 180 kg/m 2 . Equivalent length of downcomer (see Table B.7 for equivalent length of 90° bends) = 11 +

32 × 0.285 × 6 = 54.7 m. Total equivalent length = 11 + 54.7 = 65.7 m Friction loss in downcomer = 810 × 10 −6 fL e v m w 2 /d i 5 = 810 × 10 −6 × 0.012 × 66 × 0.001271 × 88.6 × 88.6/0.285 5 = 3.4 kPa = 347 kg/m 2 Boiling height is then determined. L b = CR × L × (h l −h m )/(h v −h f ) = 11 × 6(284.27 −

269.5)/(666.1 − 121.2) = 1.79 m. Note that a large boiling length reduces the CR as the net thermal head for forcing the two-phase flow through the evaporator tubes reduces.

A higher feed water temperature entering the drum helps but not close to saturation as that will hinder flow through downcomer tubes in case the downcomer tubes are partly heated.

The gravity loss in the boiling length before start of boiling = 1.79/0.001285 = 1393 kg/m 2 (an average specific volume between that of saturated liquid and the colder water entering the evaporator is taken).

The gravity loss in the evaporator section is ΔP g = 0.00981Lr 4 /v f .r 4 from Figure 2.14c =

0.65. ΔP g = 0.00981 × (6 − 1.79) × 0.65/.0013 = 20.62 kPa = 2104 kg/m 2 .

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

The friction loss may be shown to be very small due to the large number of tubes

used. r 3 = 2.3 from Figure 2.14b. ΔP f = 38 × 10 −12 fv l LG i 2 r3/d i G = 58,000 × 11 × 4/(3.14 × 420 × .044 2 ) = 999,531 kg/m 2 h

ΔP f = 38 × 10 −12 × 0.02 × 0.0013 × 4.21 × (999,531) 2 × 2.3/.044 = 0.217 kPa = 22 kg/m 2 Average mass flow in each tube = 58,000 × 11/420/3,600 = 0.42 kg/s Friction loss in the single-phase heated section = ΔP f = 810 × 10 −6 × 0.02 × 1.79 × 0.001285 × 0.42 2 /0.044 5 = 0.04 kPa = 4 kg/m 2 At entrance to evaporator bundle, velocity = 1.246 × 0.42 × 0.001271/0.044 2 = 0.34 m/s Specific volume of steam mixture at evaporator exit = 0.0909 × 0.035 + (1 − 0.0909) × 0.0013 = 0.004363 m 3 /kg (1/CR = 0.0909) Entrance loss = 0.5VH = 0.5 × 0.34 × 0.34/2/9.8/.001271 = 2.32 kg/m 2 Exit velocity = 1.246 × 0.42 × 0.004363/.044/.044 = 1.179 m/s. Exit loss = 1 VH = 1.179 ×

1.179/2/9.8/.004363 = 16.3 kg/m 2

Acceleration loss ΔP a = 7.65 × 10 −11 v f G i 2 r 2

r 2 = 1.45 from Figure 2.14a. ΔP a = 7.65 × 10 −11 × 0.0013 × (999,531) 2 × 1.45 = 0.14 kPa = 14 kg/m 2 Velocity of steam in riser pipe = 1.246 × 88.6 × 0.004363/0.317 2 = 4.79 m/s Inlet and exit loss = 1.5VH = 1.5 × 4.79 2 × 4.79/(2 × 9.8 × 0.004363) = 402 kg/m 2 L e = equivalent length = 11 + 5 × 32 × 0.317 = 61.7 m Friction loss = 810 × 10 −6 fL e v e W 2 /d i 5 = 810 × 10 −6 × 0.011 × 61.7 × 0.004363 × 88.6 × 88.6/0.317 5 = 0.095 kPa = 952 kg/m 2 (using homogeneous model)

The difference in height between the drum centerline and the evaporator header is about 2 m. The gravity head in the riser pipe = 2/0.004363 = 458 kg/m 2 . (Using the rela- tion between slip factor, void fraction, and quality, one can come up with more accurate estimation of gravity loss.)

Summary of Losses 1. Downcomer inlet and exit loss = 180 kg/m 2

2. Downcomer friction loss = 347 kg/m 2 3. Gravity loss in evaporator section = 1393 + 2104 = 3497 kg/m 2 4. Friction loss in evaporator length = 22 + 4 + 2 + 16 = 44 kg/m 2 5. Acceleration loss = 14 kg/m 2 6. Riser inlet and exit loss = 402 kg/m 2

7. Riser friction loss = 952 kg/m 2 8. Unheated riser pipe gravity head = 458 kg/m 2

9. Drum internals loss = 352 kg/m 2 (0.5 psi is a conservative estimate) Total losses = 6246 kg/m 2 . Available head = 6294 kg/m 2 . Hence CR is in the range of 11.

A computer program is helpful in performing these calculations more accurately. One can consider variation in steam generation in individual paths, differences in downcomer and riser sizes and lengths of parallel paths, effect of feed water temperature, and so on. Also this is a tedious calculation as we have to assume a CR and perform all these calculations and check if available head matches the losses. If not, another CR is assumed, and these calculations are repeated. For the earlier illustration, a CR close to the computer-calculated value was chosen. The manual calculation showed

a simplistic model for illustration. The first few rows of tubes of evaporator will be generating more steam, and hence, the CR in these tubes could be slightly lower. If risers or downcomers with different lengths are used, the flow in each could vary. One should then check the heat flux in each section and ensure that the CHF levels are not reached.

Steam Generator Furnace Design

Example 2.10

Pressure drop in two-phase flow may be estimated using the average specific volume as well as by using Thom’s method. Let us compare the difference. In a once-through

boiler, 15,000 kg/h of two-phase mixture at 105.5 kg/cm 2 a absolute pressure with exit

quality of 80% flows inside a pipe of inner diameter 50 mm. For an effective length of

50 m, compute the friction loss using Thom’s method and homogeneous model using average specific volume of mixture.

Solution

From steam tables, v g = 0.01728 m 3 /kg, v f = 0.001469 m 3 /kg

Mixture-specific volume at exit of boiler = 0.8 × 0.01728 + 0.2 × 0.001469 = 0.0141 m 3 /kg Average specific volume in the boiler = (0.001469 + 0.0141)/2 = 0.007785 m 3 /kg

w = 15,000/3,600 = 4.17 kg/s. r 3 from Figure 2.14b = 5.7

ΔP = 810 × 10 −6 × 0.02 × 50 × 4.17 2 × 0.007785/0.05 5 = 354 kPa = 3.6 kg/cm 2 [51 psi] Δ P(Thom) = 38 × 10 −12 × 0.02 × 0.001469 × (15,000 × 4/3.14 × 0.05 × 0.05) 2 × 5.7 × 50/0.05 =

371 kPa = 3.79 kg/cm 2 [54 psi] Hence, in the absence of Thom’s curves, one may estimate the two-phase friction loss using an average specific volume in the boiling section.

Example 2.11

A simplified version of circulation calculations for the boiler in Figure 2.13b is shown. The boiler is a natural gas–fired D-type boiler with external drum, downcomers, and

risers and generates 190 t/h of saturated steam at 40 kg/cm 2 a. Energy balance calcula-

tions were done, and the energy transferred in each section of furnace such as D tubes, front wall, rear wall, and the convection bank tubes was estimated using a thermal performance program. The steam generation in each section was estimated. Then, the geometric data for the circulation system, namely, the number of external downcomers, risers, and lengths of each section were inputted in a circulation program, results of which are shown in Table 2.4. The front and rear walls have a CR of about 20, while the D-tubes have a CR of about 12 and the bank tubes about 19 giving an overall average CR of 16.8.The calculations can

be more detailed with split-up of the bank tubes into several sections. However, this is only for illustrating that CR varies in each circuit depending on its resistance to flow and steam generation.

Natural circulation is adopted in small and large boilers up to a pressure of about 165 barg. Beyond this, forced circulation is required as the density difference between steam and water decreases.

Circulation Calculations