External radiation from cavities

Cavities are gas volumes located between tube banks in a back-pass section and from which thermal radiation is released to the tube banks. Cavities can exist between individual heat exchangers or between tube banks inside the individual heat exchanger. In figure 32 there is one small cavity between tube banks of the individual heat exchanger in the horizontal back-pass section, one large cavity between heat exchangers in the turn, and one small cavity between heat exchangers in the vertical back-pass section.

Figure 32. Locations of small cavities (SC) and large cavities (LC) in a back-pass section.

Cavities can be assumed to be closed by first tube rows of the adjacent heat exchangers.

For example, if the large cavity is taken under consideration, the radiation exchange situation is as it is illustrated in figure 33.

Figure 33. A large cavity in a turn.

This kind of situation can be considered to be a radiation exchange situation between four walls with participating gas if also tube rows closing the cavity are assumed to be walls. A simplified procedure can be applied to solve the situation so that different radiation ex-changes are calculated separately. Usually, temperatures of the tube rows closing the cavity are quite close to each other so it can be assumed that those do not exchange radiation with each other. Also, if the walls of the cavity are cooled, wall temperatures are usually quite close to tube row temperatures and so the radiation is not exchanged between walls and tube rows. Thus, in that situation, radiation exchanges between four walls of the cavity are not calculated but only the radiation from the gas to the walls is calculated. If all four walls have the same temperature, the situation is basically that the gas radiates to the single wall and calculation is thus similar as introduced in chapters 3.3.3 and 3.3.4. However, now there are four different wall temperatures and because absorptivity of the gas depends on the wall temperature, radiations from the gas to the different walls need to be calculated separately.

There are many kinds of equations that can be used to calculate the radiation heat transfer rate from the gas to the wall surface. Two of the equations, equations 3.37 and 3.38, were already introduced in chapter 3.3.3. Also, some other little different equations can be used, like equations 5.4 and 5.5, from which the equation 5.5 is recommended only for gas- and oil-fired boilers where flue gases do not contain solid particles (Taler & Taler 2009, 5).

𝜙 =^1+𝜀𝑠

2 𝐴_𝑠𝜎𝜀_𝑔(𝑇_𝑔⁴− 𝑇_𝑠⁴) (5.4)

𝜙 =^1+𝜀𝑠

2 𝐴_𝑠𝜎𝜀_𝑔(𝑇_𝑔⁴− (𝑇_𝑠/𝑇_𝑔)^3,6𝑇_𝑔⁴) (5.5) Consider now how much radiation arrives at the adjacent heat exchangers from the large cavity. In the calculation model, equation 3.38 is used to calculate radiation heat transfer rates from the gas to the walls as follows:

𝜙_{ℎ→𝑡1,𝑛𝑒𝑡} = ^𝜀𝑡1

𝛼𝑔1+𝜀𝑡1−𝛼𝑔1𝜀𝑡1𝐴_𝑠,𝑡1𝜎(𝜀_𝑔𝑇_ℎ⁴− 𝛼_𝑔1𝑇_𝑡1⁴) (5.6) 𝜙_{ℎ→𝑡2,𝑛𝑒𝑡} = ^𝜀𝑡2

𝛼𝑔2+𝜀𝑡2−𝛼𝑔2𝜀𝑡2𝐴_𝑠,𝑡2𝜎(𝜀_𝑔𝑇_ℎ⁴− 𝛼_𝑔2𝑇_𝑡2⁴) (5.7)

In the above equations, the absorptivity of the gas is defined as was introduced in chapters 3.3.3 and 3.3.4. As was mentioned earlier, the absorptivity of the gas depends on the wall temperature so in the above equations the absorptivity of the gas depends on that which wall the radiation is calculated to. It should also be noted that in the above equations, the area receiving the radiation is the cross-sectional area of the back-pass duct instead of the tube row surface area since the tube rows adjacent to the cavities were assumed to be walls.

If some of the walls in the cavity are refractory covered, the temperature of these walls is so high that the radiation from them to other walls need also be calculated. The radiation heat exchanges between different two wall systems are calculated separately using equation 3.60.

If both real walls in the large cavity are refractory covered, heat rates from them to the tube row walls are as follows:

𝜙_{𝑤1→𝑡1,𝑛𝑒𝑡} = 𝐹_{𝑤1→𝑡1}𝜀_𝑤1𝜀_𝑡1𝐴_𝑠,𝑤1𝜎(𝑇_𝑤1⁴ − 𝑇_𝑡1⁴)(1 − 𝛼_{𝑔+𝑝,𝑤1→𝑡1}) (5.8)

𝜙_{𝑤2→𝑡1,𝑛𝑒𝑡} = 𝐹_{𝑤2→𝑡1}𝜀_𝑤2𝜀_𝑡1𝐴_𝑠,𝑤2𝜎(𝑇_𝑤2⁴ − 𝑇_𝑡1⁴)(1 − 𝛼_{𝑔+𝑝,𝑤2→𝑡1}) (5.9)

𝜙_{𝑤1→𝑡2,𝑛𝑒𝑡} = 𝐹_{𝑤1→𝑡2}𝜀_𝑤1𝜀_𝑡2𝐴_𝑠,𝑤1𝜎(𝑇_𝑤1⁴ − 𝑇_𝑡2⁴)(1 − 𝛼_{𝑔+𝑝,𝑤1→𝑡2}) (5.10)

𝜙_{𝑤2→𝑡2,𝑛𝑒𝑡} = 𝐹_{𝑤2→𝑡2}𝜀_𝑤2𝜀_𝑡1𝐴_𝑠,𝑤2𝜎(𝑇_𝑤2⁴ − 𝑇_𝑡2⁴)(1 − 𝛼_{𝑔+𝑝,𝑤2→𝑡2}) (5.11)

In the above equations, the absorptivity of the gas is defined from equation 3.59. When all separate radiation heat rates have been calculated, total heat rates to the tube row walls can be calculated as follows:

𝜙_𝑡1 = 𝜙_{ℎ→𝑡1,𝑛𝑒𝑡}+ 𝜙_{𝑤1→𝑡1,𝑛𝑒𝑡}+ 𝜙_{𝑤2→𝑡1,𝑛𝑒𝑡} (5.12)

𝜙_𝑡2 = 𝜙_{ℎ→𝑡2,𝑛𝑒𝑡}+ 𝜙_{𝑤1→𝑡2,𝑛𝑒𝑡}+ 𝜙_{𝑤2→𝑡2,𝑛𝑒𝑡} (5.13)

Next, it must be determined how much of the radiation is absorbed by the first row of tubes and how much by the following rows. That can be defined from the diagram experimentally determined by the company where absorption fractions depend on the transverse pitch ratio.

After absorption fractions of different rows are known, external heat rates absorbed for ex-ample by the first three rows of the heat exchanger 1 are calculated as follows:

𝜙_{𝑒𝑥,𝑟1}= 𝑎_𝑓,𝑟1𝜙_𝑡1 (5.14)

𝜙_{𝑒𝑥,𝑟2}= 𝑎_𝑓,𝑟2(1 − 𝑎_𝑓,𝑟1)𝜙_𝑡1 (5.15)

𝜙_{𝑒𝑥,𝑟3}= 𝑎_𝑓,𝑟3(1 − 𝑎_𝑓,𝑟2)(1 − 𝑎_𝑓,𝑟1)𝜙_𝑡1 (5.16) where 𝑎_{𝑓,𝑟𝑗} is the absorption fraction of the jth row (-).

Another possibility for dividing the radiation for different rows could be to use a method where all radiation coming from the cavity is assumed to come from an infinite plane from where radiation goes to an infinite tube row. View factor between the infinite plane and the infinite tube row is as follows (Yuan et al. 2019, 7):

𝐹₁ = 1 − (1 − (^𝑑𝑜

𝑠𝑇)²)

1 2+^𝑑𝑜

𝑠𝑇tan⁻¹((^{𝑠𝑇2−𝑑𝑜2}

𝑑𝑜2 )

1

2) (5.17)

So, the proportion 𝐹₁ of the total external radiation is absorbed by the first tube row and portion 1 − 𝐹₁ goes through. View factors for other tube rows and so also proportions of the total radiation absorbed by those rows can be calculated from the following equation (Yuan et al. 2019, 7):

𝐹_𝑛 = (1 − 𝐹₁)^𝑛−1𝐹₁ (5.18)

For example, the external heat rate absorbed by the third tube row of the heat exchanger 1 is calculated as follows:

𝜙_{𝑒𝑥,𝑟3}= (1 − 𝐹₁)³⁻¹𝐹₁𝜙_𝑡1 (5.19)

The diagram that illustrates the second method is shown in figure 34.

Figure 34. Absorption fractions of different tube rows.

Some of the radiation goes totally through the heat exchanger to the second heat exchanger behind the first one. Radiation exchange needs to be considered to the second downstream heat exchanger because its temperature is low enough for the radiation exchange, but in the upstream direction, it can be considered that the second heat exchanger is in so high temper-ature that it does not exchange radiation with the cavity. Consider how the first row of the second downstream heat exchanger receives gas radiation from the large cavity. The situa-tion is illustrated in figure 35.

Figure 35. Radiation from a large cavity to a second downstream heat exchanger.

Once again, the first tube row of the heat exchanger is assumed to be a wall. For the calcu-lations, the situation is thought of as that the first tube row wall of the second downstream heat exchanger is moved to the place of the first tube row wall of the first downstream heat exchanger. Radiation heat rate from the gas of the cavity to the second downstream heat exchanger is calculated as follows:

𝜙_{ℎ→𝑡1,𝑛𝑒𝑡} = (1 − 𝑎_{𝑓,𝑡𝑜𝑡1})

⋅ 𝐴_𝑠,𝑡1 ^𝜀𝑡1

𝛼𝑔1+𝜀𝑡1−𝛼𝑔1𝜀𝑡1𝜎(𝜀_𝑔𝑇_ℎ⁴− 𝛼_𝑔1𝑇_𝑡1⁴) (5.20) where 𝛼_{𝑓,𝑡𝑜𝑡1} is the total absorption fraction of the first downstream heat exchanger (-).

Radiation heat rates from refractory walls to the second downstream heat exchanger are cal-culated as follows:

𝜙_{𝑤1→𝑡1,𝑛𝑒𝑡} = (1 − 𝑎_{𝑓,𝑡𝑜𝑡1})𝐹_{𝑤1→𝑡1}𝜀_𝑤1𝜀_𝑡1𝐴_𝑠,𝑤1𝜎(𝑇_𝑤1⁴ − 𝑇_𝑡1⁴)

⋅ (1 − 𝛼_{𝑔+𝑝,𝑤1→𝑡1}) (5.21)

𝜙_{𝑤2→𝑡1,𝑛𝑒𝑡} = (1 − 𝑎_{𝑓,𝑡𝑜𝑡1})𝐹_{𝑤2→𝑡1}𝜀_𝑤2𝜀_𝑡1𝐴_𝑠,𝑤2𝜎(𝑇_𝑤2⁴ − 𝑇_𝑡1⁴)

⋅ (1 − 𝛼_{𝑔+𝑝,𝑤2→𝑡1}) (5.22)

As can be seen from the previous equations, the amount of radiation passing through the first heat exchanger is taken into account by a factor (1 − 𝑎_{𝑓,𝑡𝑜𝑡1}). After the total heat rate to the second heat exchanger is calculated, the radiation is divided into different rows as usual.

The radiation from small cavities that may exist between individual heat exchangers or be-tween tube banks inside the individual heat exchanger is calculated similarly to the radiation from large cavities. The only difference is that the radiation is calculated only to one tube bank in both directions because the amount of radiation from those small cavities is not so remarkable as from large cavities.

After the external heat rates to the tube rows are calculated, those can be changed to the form of heat transfer coefficient which are needed in the following calculations. Coefficients for each tube row are calculated as follows using average temperatures of a particular tube row:

ℎ_𝑒𝑥 = ^𝜙𝑒𝑥

𝐴𝑠(𝑇ℎ,𝑎−𝑇𝑐,𝑎) (5.23)

In the case of flue gas in tubes air preheaters, the external radiation from cavities is not considered.