Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 2 Lecture 6 Conservation Principles: Entropy Conservation Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Welcome to this web based NPTEL course on advanced transport phenomena. My name is Nagarajan. I will be your instructor for this course. I am a professor in the Department of Chemical Engineering at IIT Madras.

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II Problem Statement: Continue with atmospheric-pressure combustor problem. Calculate rate of energy extraction, , necessary to bring product gas to 1000K.

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Solution Procedure: values will be needed to calculate which appears in the energy balance below:

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Apply M-Scopic Energy Balance to Calculate – When the total stress is split into – pl and T, energy conservation equation can be written in the useful form

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… The convective term now contains the enthalpy e + (p/r). If we now neglect: KE/mass (v2/2) terms, accum. term (ss) term ( no volumetric heat addition ) body force work

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… the macroscopic energy balance equation then can be written : where and

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Neglecting

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… To complete calculation of ,we therefore need where ideal gas mixture for each stream, and : Tabulated Tabulated

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Now is very close to therefore

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Finally, i.e., Or

CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Therefore

CONSERVATION OF ENTROPY (2ND LAW OF THERMODYNAMICS) Inequality: irreversible phenomena (e.g., diffusive momentum transfer, energy transfer, mass transfer, chemical reactions) lead to entropy production (Ds > 0)

CONSERVATION OF ENTROPY (2ND LAW OF THERMODYNAMICS) CONTD… Can be restated as entropy conservation equation:

CONSERVATION OF ENTROPY (2ND LAW OF THERMODYNAMICS) CONTD…. js” = diffusion flux vector for entropy = local volumetric rate of entropy production due to all irreversible processes occurring within fluid mixture

CONSERVATION OF ENTROPY Integral conservation equation for Fixed CV: Local PDE for differential CV:

CONSERVATION OF ENTROPY CONTD… “Jump” condition for a “surface of discontinuity”:

CONSERVATION OF ENTROPY CONTD… Inequalities: Or, locally:

CONSERVATION OF ENTROPY CONTD… This implies that for any fixed macroscopic region of space & at any instant:

CONSERVATION OF ENTROPY CONTD… Local volumetric entropy production

CONSERVATION OF ENTROPY CONTD… Steady-state => dV is minimum, i.e.: is a minimum compared to all other “eligible” steady-states subject to imposed boundary conditions.

CONSERVATION OF ENTROPY CONTD… Uses: Set important constraints on otherwise possible physicochemical processes (e.g., maximum work to separate a mixture, maximum possible efficiency of heat engines, etc.)

CONSERVATION OF ENTROPY CONTD… Uses: Provide basis for numerical solutions of non-equilibrium problems within the domain of “linear irreversible thermodynamics” (principle of minimum entropy production)

CONSERVATION OF ENTROPY CONTD… Uses: Guide selection of general constitutive laws governing diffusion (of species mass, momentum, energy) in non-equilibrium chemically-reacting mixtures

CONSERVATION OF ENTROPY CONTD… Uses: Pinpoint sources of entropy production and, hence, inefficiency in proposed or actual engineering devices Provide insights useful in optimization of such devices

CONSERVATION OF ENTROPY CONTD… Illustrative Exercise: In atmospheric combustor problem, calculate net convective outflow rate of entropy from the combustor– i.e., surface integral rs v . n dA.

CONSERVATION OF ENTROPY CONTD… Solution Procedure: Calculation of (net outflow rate of entropy from M-scopic CV) Known ? ?

CONSERVATION OF ENTROPY CONTD… Note that, whereas stream 1 is pure CH4 for which obtainable from , say, JANAF Thermochemical Tables, Streams 2 and 3 are mixtures; hence, “Mixing entropy contribution”

CONSERVATION OF ENTROPY CONTD… where Equivalently , (Similarly , stream 2 is a mixture , and this affects calculation of S 2.)

MATERIAL DERIVATIVE FORM OF CONSERVATION PDES Material derivative (D/Dt) of any function f (x,t) is defined as: Setting f = 1/r = (specific volume), we obtain MDF of total mass conservation equation:

INCOMPRESSIBLE FLUID Dr/Dt = 0 (rate of change of density of each moving fluid parcel vanishes) Then, div (v) = 0 (local condition on v(x,t)) = volumetric rate of fluid deformation

MDF OF SPECIES & ELEMENT CONSERVATION EQUATIONS Only an inflow by diffusion and/ or local chemical reaction can cause local species mass fraction wi (hence wi /mi) to change for each moving fluid parcel.

MDF OF SPECIES & ELEMENT CONSERVATION EQUATIONS CONTD… Chemical reactions are incapable of causing element mass fractions to change within each moving fluid parcel, hence along any streamline in steady flow

MDF OF SPECIES & ELEMENT CONSERVATION EQUATIONS CONTD… Linear momentum conservation: Set f = v (differences in contact stresses and/ or net body forces cause velocity changes (magnitude and/ or direction) of a moving fluid parcel)

MDF OF MOMENTUM, ENTERGY & ENTROPY CONSERVATION EQUATIONS Energy conservation: Set f = e + v2/2 Entropy conservation: (f = s) 34