Question

In: Other

1. Consider a simplified one-dimensional laminar flame, such as that discussed in the classroom. The earliest...

1. Consider a simplified one-dimensional laminar flame, such as that discussed in the classroom. The earliest description of a laminar flame is that of Mallard and Le Chatelier in 1883. Assume that:

a. 1-D, constant area, steady flow,

b. kinetic and potential energies, viscous shear work, and thermal

radiation are neglected,

c. pressure is constant,

d. diffusion of heat and mass are governed by Fourier’s and Fick’s laws

and binary diffusion is applied,

e. Lewis number is unity,

f. individual specific heats are all equal and constant,

g. fuel and oxidizer form products in a single-step exothermic reaction,

h. fuel is completely consumed at the flame with oxidizer in

stoichiometric or excess proportion.

By applying suitable boundary conditions, please derive the expressions for laminar flame velocity and flame thickness. (Simplified solution can be found in most of the combustion textbook, such as the one by S.R. Turns(1996) or the one by I. Glassman (1993 or newer edition).)

Solutions

Expert Solution

1) Mass Conservation equation

Hence,

is the mass flux

Species conservation equation

     

Change in mass flux is balanced by change in mass due to chemical reaction

is the mass fraction of species, is mass diffusivity, is mass produced due to chemical reaction

We balance the stoichiometric equation

1 units of fuel is mixed with units of oxidizer to give of products

Hence,

         

Energy conservation equation

                ----------------------(1)

where is the heat produced due the exothermal equation and is the sum of the heat capacity of individual species

Hence,

      

Since, we can take out as common and club all other individual heat capacities

Now, Lewis number is 1 which means mass diffusivity to heat diffusivity is same or

Putting above relation in eq (1), we get

                                            ----------------------(2)

The flame velocity can be calculated from its mass flux by

                 where S_c is flame velocity

Boundary conditions

Suppose the flame has a thickness

Upstream                                                                                   Downstream

                                                            

                                                                

Integrating equation (2), we get

now converting the spatial variable in integration to temperature variable

Now we define the average reaction rate or temperature averaged mass production as

Hence, the energy conservation equation becomes

               --------------------(3)

Above equation has two unknowns, so we have to make an assumption here. We assume that the rate of reaction is slow in upstream than in downstream, then original energy equation becomes

Putting \delta in eq (3)

We know flame speed and

flame thickness

From , we can calculate both flame speed and flame thickness


Related Solutions

1. A premixed stoichiometric methane-air flame has a laminar flame speed of 0.33 m/s and a...
1. A premixed stoichiometric methane-air flame has a laminar flame speed of 0.33 m/s and a temperature of 2200 K. The reactants are initially at 300 K and 1 atm. Find the velocity of the combustion products relative to the flame front and the pressure change across the flame. Assume that the reaction goes to completion and there is no dissociation. 2. Repeat Problem 1 for a premixed stoichiometric methanol-air flame. The flame speed is 0.48 m/s and the flame...
1. A premixed stoichiometric methane-air flame has a laminar flame speed of 0.33 m/s and a...
1. A premixed stoichiometric methane-air flame has a laminar flame speed of 0.33 m/s and a temperature of 2200 K. The reactants are initially at 300 K and 1 atm. Find the velocity of the combustion products relative to the flame front and the pressure change across the flame. Assume that the reaction goes to completion and there is no dissociation. 2. Repeat Problem 1 for a premixed stoichiometric methanol-air flame. The flame speed is 0.48 m/s and the flame...
Consider the two-dimensional laminar boundary layer flow of air over a wide 15 cm long flat...
Consider the two-dimensional laminar boundary layer flow of air over a wide 15 cm long flat plate whose surface temperature varies linearly from 20°C at the leading edge to 40°C at the trailing edge. This plate is placed in an airstream with a velocity of 2 m/s and a temperature of 10°C. Numerically determine how the surface heat flux varies along the plate. Explain the results obtained with proper reasoning. Support your results with scaling analysis wherever possible.I need c++...
Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P...
Consider the Hamiltonian of a particle in one-dimensional problem defined by: H = 1 2m P 2 + V (X) where X and P are the position and linear momentum operators, and they satisfy the commutation relation: [X, P] = i¯h The eigenvectors of H are denoted by |φn >; where n is a discrete index H|φn >= En|φn > (a) Show that < φn|P|φm >= α < φn|X|φm > and find α. Hint: Consider the commutator [X, H] (b)...
Consider a particle that is confined by a one dimensional quadratic (harmonic) potential of the form...
Consider a particle that is confined by a one dimensional quadratic (harmonic) potential of the form U(x) = Ax2 (where A is a positive real number). a) What is the Hamiltonian of the particle (expressed as a function of velocity v and x)? b) What is the average kinetic energy of the particle (expressed as a function of T)? c) Use the Virial Theorem (Eq. 1.46) to obtain the average potential energy of the particle. d) What would the average...
Consider a one-dimensional lattice with a basis of two non-equivalent atoms of masses M_1and M_2. 1....
Consider a one-dimensional lattice with a basis of two non-equivalent atoms of masses M_1and M_2. 1. Find the dispersion relations (ω versus k). 2. Sketch the normal modes in the first Brillouin zone. 3. Show that the ratio of the displacements of the two atoms u/v for the k = 0 optical mode is given by:u/v=-M_2/M_1
Consider a particle of mass m confined to a one-dimensional box of length L and in...
Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction. For a partide in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why < px2>= n2h2/4L2
Consider the one-dimensional motion of a particle of mass ? in a space where uniform gravitational...
Consider the one-dimensional motion of a particle of mass ? in a space where uniform gravitational acceleration ? exists. Take the vertical axis ?. Using Heisenberg's equation of motion, find the position-dependent operators ? ? and momentum arithmetic operator ?? in Heisenberg display that depend on time. In addition, calculate ??, ?0, ??, ?0.
Consider a particle of mass m that can move in a one-dimensional box of size L...
Consider a particle of mass m that can move in a one-dimensional box of size L with the edges of the box at x=0 and x = L. The potential is zero inside the box and infinite outside. You may need the following integrals: ∫ 0 1 d y sin ⁡ ( n π y ) 2 = 1 / 2 ,  for all integer  n ∫ 0 1 d y sin ⁡ ( n π y ) 2 y = 1...
Consider the one dimensional model of one-particle-in-a-box. Under what condition the two quantum levels are orthogonal....
Consider the one dimensional model of one-particle-in-a-box. Under what condition the two quantum levels are orthogonal. Namely, find the relation between m and n so that < m | n > = 0
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT