Outside convective thermal resistance

Let us define the outside convective thermal resistance for the tube row that is part of the tube bank. The uncorrected Nusselt number for the tube row can be calculated using several different correlations found in the literature. In the calculation model, the following Gniel-inski correlation is used (VDI 2010, 725):

𝑁𝑢_{𝑜,𝑢𝑛𝑐} = 0,3 + √𝑁𝑢_𝑙𝑎𝑚² + 𝑁𝑢_{𝑡𝑢𝑟𝑏}² (5.33)

where 𝑁𝑢_{𝑢𝑛,𝑐} is the uncorrected outside Nusselt number (-), 𝑁𝑢_𝑙𝑎𝑚 is the laminar part of the Nusselt number (-) and 𝑁𝑢_{𝑡𝑢𝑟𝑏} is the turbulent part of the Nusselt number (-).

Laminar and turbulent parts of the Nusselt number are as follows (VDI 2010, 725):

𝑁𝑢_𝑙𝑎𝑚 = 0,664√𝑅𝑒𝜓 √𝑃𝑟³ (5.34)

𝑁𝑢_{𝑡𝑢𝑟𝑏} = ^{0,037𝑅𝑒𝜓}

0,8𝑃𝑟 1+2,443𝑅𝑒_𝜓^−0,1(𝑃𝑟 2 3−1)

(5.35)

The above equations are valid if the Reynolds number is in a range from 10 to 10⁶ and the Prandtl number is in a range from 0,6 to 10³ (VDI 2010, 725). The Reynolds number 𝑅𝑒_𝜓 is defined as follows:

𝑅𝑒_𝜓 = (1 − _𝑠𝑇¹

𝑑𝑜

) ⋅^{𝑅𝑒⋅𝜋}

2⋅𝜓 (5.36)

where 𝜓 is the void fraction (-).

Reynolds number 𝑅𝑒 is calculated as follows:

𝑅𝑒 = ^{𝑞𝑚⋅𝑑𝑜}

𝐴_{𝑠,𝑒𝑓𝑓}⋅𝜇 (5.37)

where 𝐴_{𝑠,𝑒𝑓𝑓} is the effective surface area (m²).

Effective surface area is calculated as follows:

𝐴_𝑒𝑓𝑓 = 𝑊 ⋅ 𝐷 − 𝑑_𝑜⋅ 𝐷 ⋅ 𝑁𝑡𝑢𝑏𝑒𝑠 𝑖𝑛 𝑟𝑜𝑤 (5.38)

Void fraction is obtained from the equation 5.39 or 5.40 depending on the longitudinal pitch ratio as follows (VDI 2010, 726):

𝜓 = 1 − ^𝜋

4𝑎 𝑏 ≥ 1 (5.39)

𝜓 = 1 − ^𝜋

4𝑎𝑏 𝑏 < 1 (5.40)

The longitudinal pitch and so also the longitudinal pitch ratio may vary in the reality if tube bank heat exchanger has many rows in the same pass which is illustrated in figure 39.

Figure 39. Different longitudinal pitches of a tube bank.

As can be seen from the above figure, the longitudinal pitch is bigger between tube rows in different passes than between tube rows in the same pass. However, differences between different pitches are usually not very big so in the calculation model, the average longitudinal pitch is used for every tube row.

Nusselt number for the tube row after the tube configuration is considered is as follows:

𝑁𝑢_𝑜= ^{2⋅𝑓𝑎𝑙𝑖𝑔/𝑠𝑡𝑎𝑔}

𝜋 ⋅ 𝑁𝑢_{𝑜,𝑢𝑛𝑐} (5.41)

where 𝑁𝑢_𝑜 is the outside Nusselt number (-) and 𝑓_{𝑎𝑙𝑖𝑔/𝑠𝑡𝑎𝑔} is the correction factor for the tube configuration (-).

Tube configuration correction factors for aligned and staggered arrangement are as follows (VDI 2010, 726):

𝑓_{𝑎𝑙𝑖𝑔} = 1 + 0,7 (𝑏/𝑎−0,3)

𝜓^1,5(𝑏/𝑎+0,7)² (5.42)

𝑓_{𝑠𝑡𝑎𝑔} = 1 + ²

3𝑏 (5.43)

Also, it affects the heat transfer that is the fluid flowing across the tube bank heated or cooled. This is considered so that the Nusselt number is multiplied by a correction factor K which is calculated for gases as follows (VDI 2010, 727-728):

𝐾_𝑜 = (^{𝑇𝑜,𝑎}

𝑇𝑠,𝑜)

𝑛

(5.44)

where 𝐾_𝑜 is the outside correction factor for heat transfer direction (-) and 𝑇_𝑠,𝑜 is the outside surface temperature (K).

Value for n is not very much studied but for situations where the air is cooled in a tube bank, value 0 for n can be found from the literature (VDI 2010, 728). That can be applied probably in good accuracy in cases of flue gas cooling and air heating which are the two possible situations in the back-pass heat exchangers.

The uncorrected outside convective heat transfer coefficient for the tube row is calculated as follows:

ℎ_{𝑜,𝑐,𝑢𝑛𝑐} =^{𝑁𝑢𝑜⋅𝑘}

𝑑_𝑜 (5.45)

where ℎ_{𝑜,𝑐,𝑢𝑛𝑐} is the uncorrected outside convective heat transfer coefficient (-).

The corrected outside convective heat transfer coefficient for the tube row is calculated as follows:

ℎ_𝑜,𝑐 = 𝑓_𝑅𝑁𝑓_𝑓𝑎ℎ_{𝑜,𝑐,𝑢𝑛𝑐} (5.46)

where ℎ_𝑜,𝑐 is the outside convective heat transfer coefficient (-), 𝑓_𝑅𝑁 is the row number cor-rection factor (-) and 𝑓_𝑓𝑎 is the flow angle correction factor (-).

The effect of turbulence increasing in a tube bank is taken into account for a particular tube row using a correction factor 𝑓_𝑅𝑁. Conventionally when the tube bank is calculated as a whole, the average outside convective heat transfer coefficient for the whole tube bank is wanted to know and the following equation is used:

ℎ_{𝑜,𝑐,𝑎} = 𝑓_𝑁𝑅𝑓_𝑓𝑎ℎ_{𝑜,𝑐,𝑢𝑛𝑐} (5.47)

where 𝑓_𝑁𝑅 is the number of rows correction factor (-).

The number of the rows correction factor is applied for the tube banks which have less than 10 tube rows and it can be calculated using the correlation experimentally determined by the company:

𝑓_{𝑁𝑟𝑜𝑤𝑠} = 𝑎 + 𝑏 ⋅ ln(𝑁_𝑅)^𝑥− 𝑐 ⋅ (ln(𝑁_𝑅))^𝑦

+ 𝑑 ⋅ (ln(𝑁_𝑅))^𝑧− 𝑒 ⋅ (ln(𝑁_𝑅))^𝑘 (5.48) where 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑥, 𝑦, 𝑧, 𝑘 are experimentally determined coefficients (-).

A similar type of correction equation cannot be found easily for correcting the local outside convective heat transfer coefficient for a certain tube row in a tube bank. However, the above equation 5.48 can be used as a foundation to create an equation for the row number correction factor. As was mentioned earlier, turbulence increases just between 1 and 5 tube rows and then stays constant, so the following modified equation derived from equation 5.48 is de-cided to use:

𝑓_𝑅𝑁 = 𝑎 + 𝑏 ⋅ ln(𝑅_𝑁+ (𝑅_𝑁− 1))^𝑥

−𝑐 ⋅ (ln(𝑅_𝑁+ (𝑅_𝑁− 1)))^𝑦 +𝑑 ⋅ (ln(𝑅_𝑁+ (𝑅_𝑁− 1)))^𝑧

−𝑒 ⋅ (ln(𝑅_𝑁+ (𝑅_𝑁− 1)))^𝑘 (5.49) where 𝑅_𝑁 is the row number of the certain tube row (-).

Basically, a first and the last value given by equation 5.48 (𝑁_𝑅 values 1 and 9) are set for tube rows 1 and 5, and for tube rows 2 - 4 every other value given by the equation 5.48 is taken in order to get smooth turbulence increasing through the first tube rows.

A single heat exchanger may consist of several tube banks, that is, there are gaps between tube banks for example for soot blowers. These gaps are usually as big as gaps between different heat exchangers. In the literature, there is no clear agreement that what is the dis-tance from the tube bank where turbulence is reduced back to the level where it was before the tube bank. If it is assumed that the gaps between different heat exchangers are enough big, then the gaps between tube banks inside the individual heat exchanger are also enough big. Thus, the row number correction is applied always for the first tube rows in a tube bank, also inside the individual heat exchanger.

The row number correction is not applied to panel superheaters because they have an aligned arrangement and a very small longitudinal pitch ratio. Thus, especially if the longitudinal pitch ratio is less than 1.2, the amount of turbulence does not increase at all or it increases just a very little when the flow progress in a tube bank. (VDI 2010, 726)

The flow angle correction is used to correct the outside convective heat transfer coefficient if the flow hits the tube row not straight but oblique, that is, the flow angle is 0 < 𝜃 < 90.

If the flow is straight (𝜃 = 90), the value of correction is one. For the flow angle correction, the following correlation experimentally determined by the company is used:

𝑓_𝑓𝑎 = 𝑎 ⋅ 𝑒^−𝑥+ 𝑏 ⋅ 𝑒^−𝑦 ⋅ 𝜃^𝑞

−𝑐 ⋅ 𝑒^−𝑧⋅ 𝜃^𝑟+ 𝑑 ⋅ 𝑒^−𝑘⋅ 𝜃^𝑠 (5.50) where 𝜃 is the flow angle (°) and 𝑎, 𝑏, 𝑐, 𝑑, 𝑞, 𝑟, 𝑠, 𝑥, 𝑦, 𝑧, 𝑘 are experimentally determined coefficients (-).

After the outside convective heat transfer coefficient is obtained, the outside convective ther-mal resistance is calculated for the outside surface area as follows:

𝑅_𝑜,𝑐 = ¹

ℎ𝑜,𝑐𝐴𝑠,𝑜 (5.51)

where 𝑅_𝑜,𝑐 is the outside convective thermal resistance (K/W) and 𝐴_𝑠,𝑜 is the outside surface area (m²).

In document Matrix heat transfer calculation model for back-pass tube bank heat exchangers of fluidized bed steam boilers (sivua 71-76)