# Negative heat transfer coefficient for a concentric tube annulus

The heat transfer in a tube annulus can be calculated from

As equation (8.73) and (8.74) plus

qi = hi(Twi-Tf) & q0 -ho(Two-Tf) combining the four equations to replace the heat flux ratio to wall and flow temperatures, then do the calculation:

For a turbulent flow at Re=12000, Pr=0.7, Di/Do≈0.92, NuiHA ≈186 and HUoHA≈189,

( ) = 0.165 and ( ) = 0.158

Twi=790 K, Two= 693 K and Tg=698 K

Then,

Nui=189 Nuo=-361

For a laminar flow, according to Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed, pp456-458, if the model assumes a gas temperature of 100_C and surface areas of unity.

For Di/Do = 0.4 Table 8.3 gives values of 6.583, 4.979, 0.603 and 0.1823 for Nuii (NuiHA), Nuoo (NuoHA), ( ) and ( ) respectively. Dh = 0.6 and k is assumed to be unity.

It can be derived that

Ti To hai hbi hi hao hbo ho

109.6 97.9 12.3267 -11.806 11.09691 9.323204 21.57272 -0.94952

109.4 98.1 12.3267 -10.6816 11.19036 9.323204 21.12328 -1.79431

109 98.5 12.3267 -8.43284 11.38972 9.323204 20.22442 -4.15974

108.2 99.2 12.3267 -4.49751 11.77823 9.323204 18.4267 -13.7102

107.9 99.5 12.3267 -2.81095 11.97089 9.323204 17.75255 -26.1819

107.8 99.6 12.3267 -2.24876 12.0384 9.323204 17.52783 -34.4964

107.6 99.8 12.3267 -1.12438 12.17876 9.323204 17.0784 -76.0688

107.5 99.9 12.3267 -0.56219 12.25174 9.323204 16.85368 -159.214

107.3 100.1 12.3267 0.562189 12.40372 9.323204 16.40425 173.3657

106.7 100.6 12.3267 3.373135 12.83016 9.323204 15.05596 34.41647

105.2 102.1 12.3267 11.80597 14.59708 9.323204 11.68522 14.8876

102.2 104.9 12.3267 27.54727 24.84819 9.323204 4.943747 10.33213

Can anyone tell me what has gone wrong here?

As equation (8.73) and (8.74) plus

qi = hi(Twi-Tf) & q0 -ho(Two-Tf) combining the four equations to replace the heat flux ratio to wall and flow temperatures, then do the calculation:

For a turbulent flow at Re=12000, Pr=0.7, Di/Do≈0.92, NuiHA ≈186 and HUoHA≈189,

( ) = 0.165 and ( ) = 0.158

Twi=790 K, Two= 693 K and Tg=698 K

Then,

Nui=189 Nuo=-361

For a laminar flow, according to Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed, pp456-458, if the model assumes a gas temperature of 100_C and surface areas of unity.

For Di/Do = 0.4 Table 8.3 gives values of 6.583, 4.979, 0.603 and 0.1823 for Nuii (NuiHA), Nuoo (NuoHA), ( ) and ( ) respectively. Dh = 0.6 and k is assumed to be unity.

It can be derived that

Ti To hai hbi hi hao hbo ho

109.6 97.9 12.3267 -11.806 11.09691 9.323204 21.57272 -0.94952

109.4 98.1 12.3267 -10.6816 11.19036 9.323204 21.12328 -1.79431

109 98.5 12.3267 -8.43284 11.38972 9.323204 20.22442 -4.15974

108.2 99.2 12.3267 -4.49751 11.77823 9.323204 18.4267 -13.7102

107.9 99.5 12.3267 -2.81095 11.97089 9.323204 17.75255 -26.1819

107.8 99.6 12.3267 -2.24876 12.0384 9.323204 17.52783 -34.4964

107.6 99.8 12.3267 -1.12438 12.17876 9.323204 17.0784 -76.0688

107.5 99.9 12.3267 -0.56219 12.25174 9.323204 16.85368 -159.214

107.3 100.1 12.3267 0.562189 12.40372 9.323204 16.40425 173.3657

106.7 100.6 12.3267 3.373135 12.83016 9.323204 15.05596 34.41647

105.2 102.1 12.3267 11.80597 14.59708 9.323204 11.68522 14.8876

102.2 104.9 12.3267 27.54727 24.84819 9.323204 4.943747 10.33213

Can anyone tell me what has gone wrong here?