Heat flow analogy

Both the heat flow analogy and the NMR experiment can be represented by a pair of linked differential equations ... 

Figure 5.26 isobars for two-dimensional isothermal flow into a mould, computed using a thermal heat flow analogy. 

Another design option that can be considered if a column will not fit is use of an intermediate reboiler or condenser. An intermediate condenser is illustrated in Fig. 14.5. The shape of the box is now altered because the intermediate condenser changes the heat flow through the column. The particular design shown in Fig. 14.5 would require that at least part of the heat rejected from the intermediate condenser be passed to the process. An analogous approach can be used to evaluate the possibilities for use of intermediate reboilers. Flower and Jackson," Kayihan, and Dhole and Linnhofl have presented procedures for the location of intermediate reboilers and condensers. 

Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

A general method of estimating the temperature distribution in a body of any shape consists of replacing the heat flow problem by the analogous electrical situation and measuring the electrical potentials at various points. The heat capacity per unit volume C.,p is represented by an electrical capacitance, and the thermal conductivity k by an... 

Drawing an analogy with the case of heat flow by conduction through a three-layered wall discussed earlier, one can write... 

No and Kazimi (1982) derived the wall heat transfer coefficient for the forced-convective two-phase flow of sodium by using the momentum-heat transfer analogy and a logarithmic velocity distribution in the liquid film. The final form of their correlation is expressed in terms of the Nusselt number based on the bulk liquid temperature, Nuft ... 

The mass and heat transfer analogies make possible an evaluation of the mass-transfer coefficient (k) and provide insight into how membrane geometry and fluid-flow conditions can be specified to optimize flux (4). For laminar flow ... 

Time is readily sensed physiologically, and the so-called flow of time is measured by the regularity of motions and changes occurring within and around us. Plato s interpretation of Heraclitus analogy of fife to a river is that all things are in flux. We are immersed in a universe of processes that act as clocks, and even in solitude, we can still sense heartbeat, pulse, heat flow and the motions of other internal processes. Consciousness itself appears... 

For the corresponding energy balance in a CSTR we write an analogous expression [accumulation of heat] = [heat flow in] — [heat flow out]... 

The physics of thermal conduction and storage are, in fact, directly analogous to those of groundwater flow. Thermal conductivity (kT) and hydraulic conductivity (k) are analogous, as are heat capacity and storage coefficient and temperature (7) and hydraulic head (h). Indeed, heat flow (H) is estimated by an analogous equation to Darcy s Law ... 

The treatment of nonsteady-state diffusion is a question of solving Fick s second law of diffusion. In many cases, however, the equations can be taken from the treatments of the analogous problems in heat flow in solids. The point is that heat flow and diffusion are described by mathematically similar methods. 

Chemical potential is analogous to the temperature gradient that drives heat flow or the cell emf potential that drives electrical current flow, in that it provides the driving force for diffusive migration of chemical species from one region of the system to another. 

Solution. The amount of heat added (per unit area) at x = 0 in time dt is Pdt. Using the analogy between problems of mass diffusion and heat flow (Section 4.1), each added amount of heat, P dt, spreads according to the one-dimensional solution for mass diffusion from a planar source in Table 5.1 ... 

Biot and Daughaday (B6) have improved an earlier application by Citron (C5) of the variational formulation given originally by Biot for the heat conduction problem which is exactly analogous to the classical dynamical scheme. In particular, a thermal potential V, a dissipation function D, and generalized thermal force Qi are defined which satisfy the Lagrangian heat flow equation... 

This definition of thermal diffusivity gives the impression that it is simply a mathematical factor but, as Hands points out10, it is the parameter that relates heat flow to the energy gradient, analogous to thermal conductivity relating heat flow to the temperature gradient. 

We consider turbulent motion in a closed region of size L at whose boundaries vn = 0 (the index n is the normal to the surface S of the region) in the presence of external fields. It follows from (7) that the quantity ht = (grad a)n should be continuous at the boundary. From the analogy between (7) and turbulent heat conduction, noting that r plays the role of molecular thermal conductivity and considering the flow a as a heat flow, we find... 

OJohn J. Fortman, "Pictorial Analogies III Heat Flow, Thermodynamics, and Entropy," /. Chem. Educ., Vol. 70,1993,102-103. 

Figure 8.4 shows pictorially the analogy between the transshipment model and the heat exchanger network. The nodes on the left indicate the sources while the nodes on the right denote the destinations. The intermediate nodes, shown as boxes, are the warehouses. The simple arrows denote the heat flow from sources to warehouses and from the warehouses to destinations, while the highlighted arrows denote the heat flow from one warehouse to the one immediately below. 

Fig. 8.2 Main heat-flow rates that have to be considered in heat-flow, heat-balance and power-compensation reaction calorimeters running under strictly isothermal conditions [4]. The heat-flow rates inside a Peltier calorimeter are analogous (compare with Fig. 8.1). The direction of the heat-flow arrows corresponds to a positive heat-flow rate. For explanation of the different heat-flow rates, see the text.

It is assumed that e i ec and es ec. With these conditions, the equivalent thermal resistance is approximatively equal to the thermal resistance of the activated carbon. Therefore, the equivalent thermal conductivity along the radial direction is considered as equal to the activated carbon conductivity (Xr Xj. Along the axial direction, the thermal conductivity, Xy, is assumed to be the same as the aluminum conductivity. This condition is deduced from the electrical analog used to represent the heat flow inside the DLC by the parallel thermal resistances as follows ... 

---