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Flash vaporization balance - ODEs, deviation variables, linearization

  1. Feb 16, 2013 #1
    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 [itex]\dot{m}[/itex][itex]_{i}[/itex], [itex]\dot{m}[/itex][itex]_{l}[/itex],[itex]\dot{m}[/itex][itex]_{v}[/itex], [itex]\rho[/itex][itex]_{i}[/itex], [itex]\rho[/itex][itex]_{l}[/itex], [itex]\rho[/itex][itex]_{v}[/itex], and A.

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

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

    If the operating limit, [itex]\Delta[/itex], is 10 cm. What is the maximal change in [itex]\dot{m}_i[/itex], 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: [itex]\dot{m}[/itex][itex]_i, T_{in}, P_{in}, T_{out}, P_{out}, 1-(H_i-H_l)/(H_v-H_l), A, \rho_l, C_v [/itex]

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


    2. Relevant equations

    [itex]\dot{m}[/itex][itex]_{i}[/itex]-[itex]\dot{m}[/itex][itex]_{l}[/itex]-[itex]\dot{m}[/itex][itex]_{v}[/itex]=[itex]\dot{m}[/itex][itex]_{acc}[/itex]

    [itex]m= \rho V = \rho Ah(t)[/itex]

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


    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

    [itex]\dot{m}[/itex][itex]_{i}[/itex]-[itex]\dot{m}[/itex][itex]_{l}[/itex]-[itex]\dot{m}[/itex][itex]_{v}[/itex]=[itex]\rho[/itex][itex]_{l}Ah(t)[/itex]

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

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

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

    [itex]\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[/itex]

    The only thing I see that I can simplify is saying that the mass fraction of liquid left unvaporized is [itex]1-(H_i-H_l)/(H_v-H_l)[/itex]. 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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