Entropy production due to heat flow

Equation above shows the three contributions to the rate of entropy production due to heat flow, mass flow, and the chemical reaction, respectively, and excludes the viscous and electrical effects. As the membrane is assumed to be an isotropic medium, there will be no coupling between the vectorial heat and mass flows and scalar chemical reaction, according to the Curie-Prigogine principle. Under these conditions, entropy production equation identifies the conjugate forces and flows, and linear relations for coupled heat and mass flows become... 

Figure 3.9 Entropy production due to heat flow. The irreversible flow of heat between parts of unequal temperature results in the increase in entropy. The rate at which entropy is produced, P = djS/dt, is given by (3.5.3)...

As a second example, let us consider stability to thermal fluctuations. Let the temperature of a local region of interest be Teq + a, where Teq is the equilibrium temperature and a is a small deviation. As we have seen in Chapter 3, the entropy production due to heat flow is... 

Let us consider a system of length L in contact with a hot thermal reservoir, at a temperature Th. at one end and a cold thermal reservoirs, at temperature T, at the other (Fig. 17.1). In section 3.5, and in more detail in Chapter 16, we discussed the entropy production due to heat flow but we did not consider entropy balance in detail. We assume here that the conduction of heat is the only irreversible process. For this system, using Table 15.1 for the flows and forces, we see that the entropy production per unit volume is... 

Example 4.5 Entropy production in a flow through an annular packed bed The introduction of suitable packing into a fluid flow passage considerably enhances wall-to-fluid heat transfer, and hence reduces the entropy production due to heat transfer but increases the entropy production due to fluid-flow friction. Heat transfer to a fluid flowing in an annulus has a technical importance because we can heat or cool either or both of the surfaces independently. Entropy production provides a new criterion in analyzing such processes. In terms of the velocity and temperature profiles, the local rate of entropy production per unit volume of an incompressible Newtonian fluid for a two-dimensional annular flow is... 

As in the case of entropy production due to heat conduction, the entropy production due to a chemical reaction is a product of a thermodynamic force A/T, and a thermodynamic f[ow d /dt. The flow in this case is the conversion of... 

If no current flows (d /dt) = J2 = 0 and hence the first term is interpreted as the entropy production due to thermal conduction driven by the temperature gradient defined be T. In the absence of a temperature gradient however, T = 0 and hence the second term represents entropy production due to resistive heating, which is proportional to the square of the current. Since the sum of these two terms is known experimentally to account for the total entropy production in the process, it is inferred that L12 = L21. 

Therefore, the total entropy produced within the system must be discharged across the boundary at stationary state. For a system at stationary state, boundary conditions do not change with time. Consequently, a nonequilibrium stationary state is not possible for an isolated system for which deS/dt = 0. Also, a steady state cannot be maintained in an adiabatic system in which irreversible processes are occurring, since the entropy produced cannot be discharged, as an adiabatic system cannot exchange heat with its surroundings. In equilibrium, all the terms in Eq. (3.48) vanish because of the absence of both entropy flow across the system boundaries and entropy production due to irreversible processes, and we have dJS/dt = d dt = dS/dt = 0. 

From the individual entropy production terms, let us give the explicit expression of the term due to heat flow it reads... 

The interaction between heat and matter flows produces two effects, the Soret effect and the Dufour effect. In the Soret effect, heat flow drives a flow of matter. In the Dufour effect, concentration gradients drive a heat flow. The reciprocal relations in this context can be obtained by writing the entropy production due to diffusion and heat flow ... 

The general entropy balance relations for a control volume are given in terms of the rate of entropy change due to the heat transfer, mass flow, and entropy production... 

Due to the flow of heat from the hot part to the cold part, the temperatures eventually become equal, and the entropy production ceases. This is the state of equilibrium. The entropy production must vanish in the state of equilibrium, which implies that the force F and the corresponding flux Jq both vanish. In fact, we can deduce the properties of the equilibrium state by stipulating that all entropy production must vanish in that state. 

Note that for confined flow, the appearance of a critical Reynolds number or diameter for ReD < ReD opu the contribution to the rate of entropy production of heat transfer is large in comparison with the irreversibilities due to speed o q > Q y. The situation is reversed for ReD > ReD opt this case, viscosity losses predominate o > b q. This indicates that the rate or irreversible entropy production in the frictional losses (defining the drop in pressure) increase more rapidly with speed than the losses due to heat transfer by convection. 

Irreversible thermodynamics thus accomplishes two things. Firstly, the entropy production rate EE t allows the appropriate thermodynamic forces X, to be deduced if we start with well defined fluxes (eg., T-VijifT) for the isobaric transport of species i, or (IZT)- VT for heat flow). Secondly, through the Onsager relations, the number of transport coefficients can be reduced in a system of n fluxes to l/2-( - 1 )-n. Finally, it should be pointed out that reacting solids are (due to the... 

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. 

Entropy production for a non-equilibrium close to equilibrium is estimated with the help of Gibbs equation with the objective to estimate internal entropy production a = dj5 /df which is needed for characterization of fluxes J and forces X since as we shall later that cr can be expressed as sum of product of fluxes and forces. To illustrate this point, we consider a discontinuous system involving two chambers separated by a barrier but maintained at different temperatures Tj and T. In the present case, heat flow only occurs on account of force generated due to temperature difference (Fig. 2.4). 

It can be seen from Fig. 10.8b that the hydraulic (pressure) dynamics is much faster than the thermal (temperature) dynamics. Although the entropy flow due to the ohmic losses and the reaction is directly proportional to the current, the entropy flow due to the activation and concentration losses depends upon the gas species partial pressures. Therefore, the change in the heat production does not happen... 

As a first illustration of the theory presented in the last two sections, let us consider thermoelectric effects which involve the flow of heat Jq and electric current Ig in conducting wires (in which the subscript indicates that the flow corresponds to the flow of electrons). The entropy production per unit volume due to these two irreversible processes and the Unear phenomenological laws associated with it are... 

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