Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT
Madras Advanced Transport Phenomena Module 2 Lecture 5 Conservation
Principles: Momentum & Energy Conservation Slide 2 MASS
CONSERVATION: ILLUSTRATIVE EXERCISE Atmospheric-pressure combustor
Slide 3 MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD Slide 4
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 Slide 5 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 Slide 6
Slide 7 Slide 8 Slide 9 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 Slide 10 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 Slide 11 (See following matrix)Sought
For steady-state, this can be written as MASS CONSERVATION:
ILLUSTRATIVE EXERCISE CONTD Slide 12 Slide 13 Note that we need We
readily find; MASS CONSERVATION: ILLUSTRATIVE EXERCISE CONTD Slide
14 Slide 15 Slide 16 Just calculated MatrixSought MASS
CONSERVATION: ILLUSTRATIVE EXERCISE CONTD Slide 17 Slide 18 This
completes the composition calculation for the exit stream (Note:
(no unburned methane) MASS CONSERVATION: ILLUSTRATIVE EXERCISE
CONTD Slide 19 MOMENTUM CONSERVATION Linear Momentum Conservation
Angular Momentum Conservation Slide 20 LINEAR MOMENTUM CONSERVATION
Diffusion & Source Terms: Slide 21 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
Slide 22 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) Slide 23 v v = local, instantaneous, convective
momentum flux Tensor (as is ) requires 9 local scalar quantities
for complete specification LINEAR MOMENTUM CONSERVATION CONTD Slide
24 In cylindrical polar coordinates, components are: Because of
symmetry, only 6 are independent LINEAR MOMENTUM CONSERVATION CONTD
Slide 25 z-component of PDE: where (analogous expression can be
written for [div ] z Jump condition across surface of
discontinuity: LINEAR MOMENTUM CONSERVATION CONTD Slide 26 Slide 27
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 Slide 28 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) Slide 29 CONSERVATION OF
ENERGY Definition of Terms: Slide 30 Work Terms: CONSERVATION OF
ENERGY CONTD Slide 31 Integral Conservation Equation for Fixed CV:
where e = specific internal energy of mixture (function of local
thermodynamic state), including chemical contributions Slide 32 v 2
/2 = specific kinetic energy possessed by each unit mass of mixture
as a consequence of its ordered motion CONSERVATION OF ENERGY CONTD
Slide 33 Local PDE for Differential CV: Jump condition for surface
of discontinuity: CONSERVATION OF ENERGY CONTD Slide 34 When body
forces g i (per iunit mass) are same for all chemical species
(e.g., gravity): With the constraints: CONSERVATION OF ENERGY CONTD
Slide 35 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 Slide 36 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 ) Slide 37 c. The energy required to raise the
temperature of one gram of liquid water from 273.2 K to 373.2 K. d.
The energy required to melt one gram of ice at 273.2 K.
CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD Slide 38 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 273.2 K CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD
Slide 39 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? Slide 40 Solution Procedure: a) KE/mass for 1 g H 2 O @
1 m/s 1 m/s 1 m/s m=1g Z=0 m Z=1 m CONSERVATION OF ENERGY:
ILLUSTRATIVE EXERCISE I CONTD Slide 41 H 2 O( l ) b. c.
CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD Slide 42 d.
e. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD Slide 43
f. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE I CONTD Slide
44
