Inside convective thermal resistance

For turbulent flows inside the tubes, the Nusselt number is basically independent of surface condition and it can be calculated for both gas and liquid flows for example using the fol-lowing empirical correlation proposed by Gnielinski (VDI 2010, 696):

𝑁𝑢_𝑖 = ^{(𝑓/8)𝑅𝑒𝑃𝑟}

1+12,7√𝑓/8(𝑃𝑟^2/3−1)(1 + 𝑑_𝑖/𝑙)^2/3) (5.55) where 𝑁𝑢_𝑖 is the inside Nusselt number (-), 𝑓 is the friction factor (-) and 𝑑_𝑖 is the tube inside diameter (m).

The above equation is valid if the Reynolds number is in a range from 10⁴ to 10⁶, the Prandtl number is in a range from 0,1 to 1000 and the ratio of the tube inside diameter and tube length is less than or equal to 1. The friction factor is calculated as follows (VDI 2010, 696):

𝑓 = (1,8 log₁₀𝑅𝑒 − 1,5)⁻² (5.56)

Reynolds number is calculated as follows (VDI 2010, 696):

𝑅𝑒 =^𝑤𝑑𝑖

𝑣 (5.57)

where 𝑣 is the kinematic viscosity (m²/s).

As was mentioned in chapter 3.2, the convective heat transfer coefficient is first big and then it starts to decrease when boundary layers develop on some surface. In equation 5.55 this is considered by a term (1 + 𝑑_𝑖/𝑙)^2/3). This effect is especially important to take into account when the tubes are short. For that reason, equation 5.55 is especially good for air preheaters shown in figure 30 where individual tubes are usually quite short. In the case of other steam boiler back-pass heat exchangers, the tubes are bent to pass through the entire heat exchanger as single tubes, so the part of the tube where boundary layers develop is quite short compar-ing the whole tube length. Thus, the Nusselt number for other heat exchangers than air pre-heaters can also be calculated using the following simple Hausen equation (Vakkilainen 2016, 137):

𝑁𝑢_𝑖 = 0,0235 ⋅ (𝑅𝑒^0,8− 230) ⋅ (1,8 ∗ Pr^0,3− 0,8) (5.58)

The above equation is valid if the Reynolds number is in a range from 2300 to 10⁷, the Prandtl number is in a range from 0,6 to 500, the tube inside diameter is much smaller than the tube length and the tube wall temperature is close to the inside fluid temperature.

As can be seen from the restrictions of equations 5.55 and 5.58, the definition for the border between laminar and turbulent flow varies depending on the equation. Therefore, it is im-portant to ensure the constraints of the particular equation. For example, if the turbulent flow is assumed to be starting from 2300 and the turbulent flow equation 5.55 is used for the flow which 𝑅𝑒 = 3000, the correct value for the Nusselt number is not obtained because the lower limit was 10⁴ for that equation.

For laminar flows which have a simpler flow structure than turbulent flows, empirical cor-relations are not used because the exact value of the Nusselt number in the fully developed region can be defined theoretically by solving the energy conservation equation. A constant value for the Nusselt number is obtained from that procedure and the value depends on the surface condition. In the case of constant surface heat flux, the value is 𝑁𝑢 = 4,36 and in the case of constant surface temperature, the value is 𝑁𝑢 = 3,66. (Incropera et al. 2007, 505-507)

In back-pass heat exchangers of the steam boilers, neither of the above conditions fully cor-responds to the reality. In steam boiler calculations the value 𝑁𝑢 = 3,64 is recommended to be used (Vakkilainen 2016, 138).

In addition to the earlier introduced laminar and turbulent flow borders 2300 and 10⁴, one definition is that the flow is turbulent when 𝑅𝑒 > 4000 (Engineering ToolBox, 2004). So, depending on the source, a theoretically solved constant laminar flow value for the Nusselt number can be considered to be used up to either 2300, 4000, or 10⁴. The Nusselt number for the transition region between laminar and fully turbulent regions can be calculated also using a procedure where Nusselt numbers are first defined for the Reynolds numbers 2300 and 10⁴ and then the Nusselt number is interpolated between those values as follows (VDI 2010, 696):

𝑁𝑢_𝑖 = (1 − 𝛾)𝑁𝑢₂₃₀₀+ 𝛾𝑁𝑢₁₀⁴ (5.59) where 𝑁𝑢₂₃₀₀ is the Nusselt number for 𝑅𝑒 = 2300 and 𝑁𝑢_1𝑜⁴ is the Nusselt number for 𝑅𝑒 = 10⁴. Value for the 𝛾 is calculated as follows:

𝛾 = ^{𝑅𝑒−2300}

10⁴−2300 (5.60)

As in the case of the outside Nusselt number, the direction of the heat transfer affects also the inside Nusselt number. In the case of gases, the correction factor K for the direction of the heat transfer is calculated as follows:

𝐾_𝑖 = (^{𝑇𝑖,𝑎}

𝑇_𝑠,𝑖)

𝑛

(5.61)

where 𝐾_𝑖 is the inside correction factor for heat transfer direction (-), 𝑇_𝑖 is the inside fluid temperature (K) and 𝑇_𝑠,𝑖 is the inside surface temperature (K).

If any gas is cooled, n gets the value 0. In the case of heating, the value of n depends on the gas. For example, the value -0,18 can be used for the superheated steam if the pressure is 21-100 bar and 0,67 < 𝑇_𝑖/𝑇_𝑤 < 1. For carbon dioxide, the value of n is 0,12 when 0,5 <

𝑇_𝑖/𝑇_𝑤 < 1. In the case of liquids, the direction of the heat transfer has no effect on the inside Nusselt number. (VDI 2010, 697)

Because the value of 𝐾_𝑖 is usually very close to one, the value one is applied for the 𝐾_𝑖 in all situations in the calculation model. After the Nusselt number is defined, the inside convec-tive heat transfer coefficient is calculated as follows (VDI 2010, 696):

ℎ_𝑖,𝑐 = ^{𝑁𝑢𝑖𝑘}

𝑑_𝑖 (5.62)

where ℎ_𝑖,𝑐 is the inside convective heat transfer coefficient (-).

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

𝑅_𝑖,𝑐 = ¹

ℎ_𝑖,𝑐𝐴_𝑠,𝑖 (5.63)

where 𝑅_𝑖,𝑐 is the inside convective thermal resistance (K/W) and 𝐴_𝑠,𝑖 is the inside surface area (m²).

In the case of evaporating heat exchangers, the inside fluid boils so the inside convective heat transfer coefficient is very big (VDI 2010, 20). Therefore, the inside convective thermal resistance is very small, and it has a very small effect on the overall thermal resistance of the heat transfer system. Correlations for evaporation situations can be found in the literature.