Entropy flow rate

Ideal Gas Mixing. When C species are mixed at constant pressure and temperature, as illustrated in Figure 9.9. the change in the entropy flow rate is given by Eg. 19.10). applied separately for each species j ... 

For the open system of Fig. 2.16, one must express the entropy-flow rate as written in Eq. (2) of Fig. 2.81. Besides the entropy change due to the heat flux, there is an entropy change due to the flux of matter. The matter flux of component i is expressed by its flow-rate term, d ni/dt. The matter flux can be measured, for example, by thermogravimetry as listed in Fig. 2.16. Each flux of matter across the boundary of the system, deO, needs to be multiplied with the partial molar change in the corresponding entropy, S, [= (dS/dnil p, see Fig. 2.19], to obtain the total contribution to d S/dt. The change in partial molar entropy must be known, or be measured independently by heat capacity measurements, for example, from zero kelvin to T, as outlined in Figs. 2.22 and 2.23. 

In this section, an R-field formulation for representing the convection of a two-gas mixture is given [12, 13, 15], This is an extension of the bond graph formulation for the forced convection of a compressible ideal gas given in [6], The details of the sub-model for modelling the convection of a two-component gas mixture are given in Fig. 10.3. The most important element in the expanded model of the MR element is the RS-field element (see Fig. 10.3). This element receives the downstream side temperature and the information of the valve position (x), the upstream side chemical potentials and temperature, and the downstream side chemical potentials to calculate the mass and entropy flow rates. Note that all these variables are inputs to the MR element. To maintain the clarity of the figure, the connections needed to explicitly show these modulations are not drawn. 

The entropy flow rate associated with the mass flow rate is calculated by means of a transformer element (between junctions 1 and Im), which is modulated by the specific entropy of the upstream side gases. This information of the upstream side specific entropy either can be obtained directly from the upstream side storage element or, if a standalone scheme is required, can be calculated from the upstream side /x s and T s (which are inputs of the MR element) as... 

The entropy flow rate from the upstream side is given as 5u = msu- The R-field represents the change in the intensive variables between the upstream and the downstream sides. The temperatures, pressures and the chemical potentials of the gas mixture in the upstream and the downstream sides are imposed by the storage elements on the corresponding sides. Due to this, there is an enthalpy difference between the upstream and the downstream sides, which can be represented as the relation between the changes in the intensive variables by using the Gibbs-Duhem equation [5] as... 

RS-element (see (10.17)). The downstream side entropy flow rate is the sum of the upstream side entropy flow rate (5u imposed at 1 junction by the MTF element) and the irreversible entropy generated (5gen in (10.21)). This sum is realised by means of the zero junction shown in Fig. 10.3. 

The decrease in the current also results in the decrease in the polarisation losses and the decrease in the reaction entropy flow rate (5r) (due to reduced mass flow rates) and thereby results in the fall of the system temperature. The reverse phenomena are observed (at time 2000 s in Fig. 10.8) when the load current density is increased. 

The term B in equation (1.27) is related to the longitudinal molecular diffusion in the column. It is especially important when the mobile phase is a gas. This term is a consequence of entropy, telling us that a system will tend towards the maximum degrees of freedom as demonstrated by a drop of ink that diffuses after falling into a glass of water. Hence, if the flow rate is too slow, compounds being separated will mix faster than they will migrate. This is why one must never interrupt the separation process, even momentarily. 

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

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... 

In this chapter, we show that it is not so much energy that is consumed but its quality, that is, the extent to which it is available for work. The quality of heat is the well-known thermal efficiency, the Carnot factor. If quality is lost, work has been consumed and lost. Lost work can be expressed in the products of flow rates and driving forces of a process. Its relation to entropy generation is established, which will allow us later to arrive at a universal relation between lost work and the driving forces in a process. 

In this chapter, we first introduce the principles of irreversible or nonequilibrium thermodynamics as opposed to those of equilibrium thermodynamics. Then, we identify important thermodynamic forces X (the cause) and their associated flow rates / (the effect). We show how these factors are responsible for the rate with which the entropy production increases and available work decreases in a process. This gives an excellent insight into the origin of the incurred losses. We pay attention to the relation between flows and forces and the possibility of coupling of processes and its implications. 

In Section 3.3, we have shown that the entropy generation rate in the case of heat transfer in a heat exchanger is simply the product of the thermodynamic driving force X = A(l/T), the natural cause, and its effect, the resultant flow / = Q, a velocity or rate. Selected monographs on irreversible thermodynamics, see, for example, [1], show how entropy generation also has roots in other driving forces such as chemical potential differences or affinities. 

On the other hand, irreversible thermodynamics has provided us with the insight that entropy generation is related to process flow rates like those of volume, V, mass in moles, h, chemical conversion, vl h, and heat, Q, and their so-called conjugated forces A(P/T), -A(p/T), A/T, and A(l/T). Although irreversible thermodynamics does not specify the relationship between these forces X and their conjugated flow rates /, it leaves no doubt about the... 

Entropy generation rate as a function of the heat flow rate. 

When one tracer element is observed at the inlet, the mixing performance of the operation/equipment is evaluated from the viewpoint of information entropy based on the uncertainty regarding the amount of time taken by the tracer element to reach the outlet. The time the tracer is injected is set as the origin. The definite integral from zero to infinity of the RTD function of the dimensionless residence time (r(= t/T, T = V/Q, V volume of equipment, Q flow rate)) should be unity. Additionally, by considering the definition of dimensionless time, its average value is fixed at unity. Therefore, the standardized and restrictive conditions can be written as... 

From these values we must find the corresponding entropies from Fig. G.2. They are read at the vapor pressure for 80 degF of 101.37 kPa. The flow rates come from Problem 9.9 ... 

The parameters controlling the rate of entropy production in the tower are now obvious the vapor flow rate V (a function of the reflux ratio), the inlet and outlet mole (or mass) fractions, and the relationship between yA and y g (a function of the reflux ratio and the relative volatility). 

Available energy Mass flow rate Mass flux Pressure Gas Constant Entropy per unit mass Entropy flux Entropy production Time... 

The method of coordinatewise optimization was proposed for simultaneous choice of flow rates and pressure losses on the closed redundant schemes (Merenkov and Khasilev, 1985 Merenkov et al., 1992 Sumaro-kov, 1976). According to this method motion to the minimum point of the economic functional F(x, Pbr) is performed alternately along the concave (F(x)) and convex (F(Pbr)) directions. The convex problem is solved by the dynamic programming method and the concave one reduces to calculation of flow distribution. The pressure losses in this case are optimized on the tree obtained as a result of assumed flow shutoff at the end points of some branches. The concave problem is solved on the basis of entropy... 

The rate of entropy generation and the lost work for each of the individual ] of the process are calculated by Eqs. (16.1) and (16.15). Since the flow rate of methane is not given, we take 1 kg of methane entering as a basis. The rates S, W m, and Q are therefore expressed not per unit of time but per kg of entering meth The heat transfer for the compression/cooling step is calculated by an enc balance ... 

A turbine operates adiabatically with superheated steam entering at T, and P, with a mass flow rate m. The exhaust pressure is P2 and the turbine efficiency is i). For one of the following sets of operating conditions, determine the power output of the turbine and the enthalpy and entropy of the exhaust steam. 

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