By Chongbin Zhao, Bruce E. Hobbs, Alison Ord
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Additional resources for Convective and Advective Heat Transfer in Geological Systems Advances in Geophysical and Environ
51) This indicates that the vertical velocity, which is referred to as the upward throughflow, is constant throughout the whole layer. 44) yields the following equation: Pe ѨT ∗ Ѩ2 T ∗ = ∗2 . 53) where C1 and C2 are two independent constants. In order to determine C1 and C2 constants uniquely, we must use two thermal boundary conditions, one of which must be a temperature boundary. 48) yields T∗ = ∗ 1 Pe (e − ePe y ). 49) that P∗ is a function of y∗ only, hence, dP∗ ѨP∗ = = RaT T ∗ − Pe . 47): C3 = P1∗ + Pe − RaT Pe ePe − 1 Pe e .
Since the conductive thermal flux can be expressed as q0 = −λe ѨT Ѩy in this instance, an upward constant conductive thermal flux of q0 means that the temperature gradient, ѨT Ѩy = −q0 λe , is a positive constant, because the thermal conductivity of the crustal material is constant here. 16) cpρ V H ѨP q0 − λf 0 e = ρ f 0g − e Ѩy c p ρ f 0Vˆ ˆ 1−e c p ρ f 0 Vˆ λe y βT ρ f 0 g − μˆ V. 17) It is obvious that when advection is considered, the temperature, pore-fluid pressure and their gradients vary non-linearly with the depth of the crust.
2 shows the thermal effect of this heat conduction model on the distribution of the dimensionless pressure for two different cases. In this figure, the reference hydrostatic pressure is labeled as R. H. Pressure. In case 1, the temperature difference between the top and the bottom of the crust is 600 ◦ C, whereas in case 2, this temperature difference is 1000 ◦ C. H. 0 Fig. 2 Thermal effect on distribution of dimensionless pressure on the distribution of the dimensionless pressure, the reference hydrostatic pressure resulting from a constant reference density of the pore-fluid is also shown in this figure for comparison.
Convective and Advective Heat Transfer in Geological Systems Advances in Geophysical and Environ by Chongbin Zhao, Bruce E. Hobbs, Alison Ord