Method of characteristics for population balance equation

[Graham Power]

Homework Statement

The PDE: ∂n/∂t + G∂n/∂L=0

The initial condition: n(0,L)=ns

The boundary condition: n(t,0)=B/G

The parameter B and G above are dependent upon process conditions and change at each time. They can be calculated with adequate experimental data.

Homework Equations

My question is how to solve this using the method of characteristics?

The Attempt at a Solution

 

[epenguin]

Homework Statement

The PDE: ∂n/∂t + G∂n/∂L=0

The initial condition: n(0,L)=ns

The boundary condition: n(t,0)=B/G

The parameter B and G above are dependent upon process conditions and change at each time. They can be calculated with adequate experimental data.

Homework Equations

My question is how to solve this using the method of characteristics?

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

I never made much sense of the books re 'method of characteristics' but noticed that your equation is- (∂n/∂t)/(∂n/∂L) = G

and then the LHS is dL/dt at constant n, so you have a straightforward ordinary differential equation which you can solve (I gather G is a constant, but if it is a function of L and t you still probably can). So you have all the lines or curves at constant n, then you build up the entire solution from your initial/boundary conditions.

BTW
- (∂n/∂t)/(∂n/∂L) = (dL/dt)_n
should not be this big mystery - just draw a little pic of a bit of plane with 3 variables.This is your first post and you got a hint without yourself having taken the first step which is usually required, so please COME BACK when you have got the final answer at least. I would like to know if this is the Method of Characteristics myself.

 

[Graham Power]

I never made much sense of the books re 'method of characteristics' but noticed that your equation is


- (∂n/∂t)/(∂n/∂L) = G

and then the LHS is dL/dt at constant n, so you have a straightforward ordinary differential equation which you can solve (I gather G is a constant, but if it is a function of L and t you still probably can). So you have all the lines or curves at constant n, then you build up the entire solution from your initial/boundary conditions.

BTW
- (∂n/∂t)/(∂n/∂L) = (dL/dt)_n
should not be this big mystery - just draw a little pic of a bit of plane with 3 variables.


This is your first post and you got a hint without yourself having taken the first step which is usually required, so please COME BACK when you have got the final answer at least. I would like to know if this is the Method of Characteristics myself.

Thanks for the reply. I know that the characteristic equations for this PDE are:

∂n/∂s=0 (1)
∂L/∂s=G (2)
∂t/∂s=1 (3)

Solving the above:

n=n₀ (4)
t=s (5)
L=Gt+L₀ (6)

From what I can see, the above equations suggest that along a characteristic curve given by equation (6), the population density, n, at size L₀, travels along the size axis with rate of growth, G.

The initial condition, n(0,L)=ns describes the population of particles over a given size range. So the initial data I have is a number of values of population density, n, at a number of sizes.
Does the solution suggest that each initial value of population density, n corresponding to an initial size, L₀ will stay constant along equation (6) and correspond to a different size in the next time step? I hope I am clear in what I am saying here.