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Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 2 Lecture 5 Conservation Principles: Momentum & Energy Conservation

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MASS CONSERVATION: ILLUSTRATIVE EXERCISE Atmospheric-pressure combustor

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MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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Problem Statement: Pure methane gas at 300 K, 1 atm, and pure air at 300K, 1 atm, steadily flow into a combustor from which a single stream of product gas (CO 2, H 2, O 2, N 2 ) emerges at 1000 K, 1 atm. Use appropriate balance equations to determine: Mass flow rate of product stream out of combustor MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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Chemical composition of product gas mixture (expressed in mass fractions) Formulate & defend important assumptions. Treat air as having nominal composition O 2 = 0.23, N 2 = 0.73 MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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Total Mass Balance: i.e., exit stream (1000 K, 1 atm) has mass-flow rate of 21 g/s (via overall mass balance). MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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Chemical Composition of the Exit Stream, i.e., This can be found via the chemical element mass balances, i.e., MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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(See following matrix)Sought For steady-state, this can be written as MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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Note that we need We readily find; MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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Just calculated MatrixSought MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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This completes the composition calculation for the exit stream (Note: (no unburned methane) MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD…

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MOMENTUM CONSERVATION Linear Momentum Conservation Angular Momentum Conservation

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LINEAR MOMENTUM CONSERVATION Diffusion & Source Terms:

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LINEAR MOMENTUM CONSERVATION CONTD… = local stress operator . n dA = (Vector) element of surface force g i = local “body” force acting on each unit mass of species i

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LINEAR MOMENTUM CONSERVATION CONTD… Integral Conservation Equation for Fixed CV: Differential form (local PDE): Each equation equivalent to 3 scalar equations, one for each component (direction)

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v v = local, instantaneous, convective momentum flux Tensor (as is ) requires 9 local scalar quantities for complete specification LINEAR MOMENTUM CONSERVATION CONTD…

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In cylindrical polar coordinates, components are: Because of symmetry, only 6 are independent LINEAR MOMENTUM CONSERVATION CONTD…

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z-component of PDE: where (analogous expression can be written for [div ] z “Jump” condition across surface of discontinuity: LINEAR MOMENTUM CONSERVATION CONTD…

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CONSERVATION OF ENERGY (I LAW OF THERMODYNAMICS) In chemically-reacting systems, thermal, chemical & mechanical (kinetic) sources of energy must be considered Heat-addition & work must be included

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CONSERVATION OF ENERGY (I LAW OF THERMODYNAMICS) CONTD… Definitions: = total energy flux in prevailing material mixture = volumetric energy source for material mixture Typically derived from interaction with a local electromagnetic field (“photon phase”)

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CONSERVATION OF ENERGY Definition of Terms:

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Work Terms: CONSERVATION OF ENERGY CONTD…

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Integral Conservation Equation for Fixed CV: where e = specific “internal” energy of mixture (function of local thermodynamic state), including chemical contributions

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v 2 /2 = specific kinetic energy possessed by each unit mass of mixture as a consequence of its ordered motion CONSERVATION OF ENERGY CONTD…

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Local PDE for Differential CV: “Jump” condition for surface of discontinuity: CONSERVATION OF ENERGY CONTD…

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When body forces g i (per iunit mass) are same for all chemical species (e.g., gravity): With the constraints: CONSERVATION OF ENERGY CONTD…

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If g i is associated wioth a time-independent potential energy field, , then the total energy density field becomes: (separate body-force term on RHS not required) CONSERVATION OF ENERGY CONTD…

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CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I Problem Statement: Numerically, compute and compare the following energies (after converting all to the same units, say calories): a. The kinetic energy of a gram of water moving at 1 m/s b. The potential energy change associated with raising one gram of water through a vertical distance of one meter against gravity (where g=0.9807*10 3 cm/s 2 )

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c. The energy required to raise the temperature of one gram of liquid water from K to K. d. The energy required to melt one gram of ice at K. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD…

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e. The energy released when one gram of H 2 O(g) condenses at 373 K. f. The energy released when one gram of liquid water is formed from a stoichiometric mixture of hydrogen (H 2 (g)) and oxygen (O 2 (g)) at K CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD…

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What do these comparisons lead you to expect regarding the relative importance of changes of each of the above mentioned types of energy in applications of law of conservation of energy? Is H 2 O “singular,” or are you conclusions likely to be generally applicable?

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Solution Procedure: a) KE/mass for 1 g H 2 1 m/s 1 m/s 1 m/s m=1g Z=0 m Z=1 m CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD…

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H 2 O( l ) b. c. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD…

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d. e. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD…

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f. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD…

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