Heat flow equation, four-term

Including also the heat capacities of the sample and reference platforms, Cs, into the model leads to the bottom sketch in Fig. A. 11.2. The heat-flow rate into the sample needs now a four-term heat-flow-equation making use of a second temperature difference AT = Tj - T, besides AT = T - Tji... 

The physical meaning of the terms (or group of terms) in the entropy equation is not always obvious. However, the term on the LHS denotes the rate of accumulation of entropy within the control volume per unit volume. On the RHS the entropy flow terms included in show that for open systems the entropy flow consists of two parts one is the reduced heat flow the other is connected with the diffusion flows of matter jc, Secondly, the entropy production terms included in totai demonstrates that the entropy production contains four different contributions. (The third term on the RHS vanishes by use of the continuity equation, but retained for the purpose of indicating possible contributions from the interfacial mass transfer in multiphase flows, discussed later). The first term in totai arises from heat fluxes as conduction and radiation, the third from diffusion, the fourth is connected to the gradients of the velocity field, giving rise to viscous flow, and the fifth is due to chemical reactions. 

The use of the corrections B in Fig. A.l 1.2 needs two calibration runs of the DSC of Fig. 4.54. The heat capacities of the calorimeter platforms, C pi and C pi, and the resistances to the constantan body, R pi and R pi, must be evaluated as a function of temperature. First, the DSC is ran without the calorimeters, next a run is done with sapphire disks on the sample and reference platforms without calorimeter pans. From the empty run one sets a zero heat-flow rate for and This allows to calculate the temperature-dependent time constants of the DSC, written as = C piRspi and Tr = CrpiRrpi, and calculated from the equations in the lower part of Fig. A. 11.2. For the second run, the heat-flow rates are those into the sapphire disks, known to be mCpQ, as suggested in Figs. 4.54 and 4.70. The heat-capacity-correction terms are zero in this second calibration because no pans were used. From these four equations, all four platform constants can be evaluated and the DSC calibrated. 

Rearranging these equations, where Q=Q-Q, results in the following four-term heat flow expression ... 

The governing equations (9), (10) and (11) are spatially diseretized using a Galerkin finite-element approach in conjunction with P2-P1 tetrahedral Taylor-Hood elements, with ten nodes for velocity and temperature, and four nodes for pressure. A finite element solver for three-dimensional non-Newtonian fluid flow and advection-diffiision heat transfer has been developed. The unknown veloeity, pressure and temperature flelds can be expressed in terms of the shape fimctions and, and nodal... 

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