Homework Help: Flash vaporization balance - ODEs, deviation variables, linearization

1. Feb 16, 2013

halycos

1. The problem statement, all variables and given/known data

Given the attached figure,

a) Develop an ordinary differential equation that describes the dynamic height h(t) in the flash tank in terms of $\dot{m}$$_{i}$, $\dot{m}$$_{l}$,$\dot{m}$$_{v}$, $\rho$$_{i}$, $\rho$$_{l}$, $\rho$$_{v}$, and A.

b) Given the fact that the process is isenthalpic, eliminate $\dot{m}$$_{v}$ from the equation in part (a).

c) Develop a nonlinear ordinary differential equation assuming $\dot{m}_{l}=C_{v}\sqrt{h(t)}$. Simplify as much as possible.

If the operating limit, $\Delta$, is 10 cm. What is the maximal change in $\dot{m}_i$, so steady state is reached before the operating limit is reached?

d) Determine the above using steady state conditions, deviation variables, and linearization.
Values are given for the following variables: $\dot{m}$$_i, T_{in}, P_{in}, T_{out}, P_{out}, 1-(H_i-H_l)/(H_v-H_l), A, \rho_l, C_v$

e) Solve the nonlinear DE, and compare with (d)

2. Relevant equations

$\dot{m}$$_{i}$-$\dot{m}$$_{l}$-$\dot{m}$$_{v}$=$\dot{m}$$_{acc}$

$m= \rho V = \rho Ah(t)$

For isenthalpic processes, mass fraction vaporized = $Y= (H_i-H_l)/(H_v-H_l)$

3. The attempt at a solution

Were told to include the effect of vapor mass to the height of the vessel liquid, so the equation for a should be

$\dot{m}$$_{i}$-$\dot{m}$$_{l}$-$\dot{m}$$_{v}$=$\rho$$_{l}Ah(t)$

$\frac{dh(t)}{dt}$=$\frac{1}{\rho_l A}($ $\dot{m}$$_{i}$-$\dot{m}$$_{l}$-$\dot{m}$$_{v}$)

For (b), given specific enthalpies, then $\dot{m}_v=Y\dot{m}_i = Y= \dot{m}_i (H_i-H_l)/(H_v-H_l)$.

My issues start at (c). I'm able to find the following equation

$\frac{dh(t)}{dt}+C_v \sqrt{h(t)}/\rho _l A = [m_i(H_i-H_l)/(H_v-H_l)]/ \rho _l A$

The only thing I see that I can simplify is saying that the mass fraction of liquid left unvaporized is $1-(H_i-H_l)/(H_v-H_l)$. Otherwise, I don't see any way of reducing it further. Also, there are no temperature or pressure dependencies, and I am given values for these at steady state to use for parts (d) and (e). I don't even know how to apply the deviation variables without these dependencies.
If anybody can offer some input, I'd be very grateful.

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