■ Derivation of Helmholtz and Taylor Instability Criteria

- Hetsroni, "Handbook of Multiphase System"

(1) Helmholtz instability ( Lamb and Zuber )

From initially isolated bubbles, as the heat flux increases, "vapor columns" occur.

Let the velocity of surface wave propagation be C

[eqn: rm C^2 ~=~ ( n over m ) ^2 ~=~ { □ m g_c }over { rho_l _ rho _v } - { rho_l rho_v } over { ( rho_l + rho_v ) ^2 } ( V_v - V_l ) ^2]

For a harmonic wave form, the amplitude is

[eqn: rm □ ~=~ □_o`` e^-i(nt) cos (mx)]

where the wave number ; [eqn: rm m ~=~ 2pi over □]

the wave angular velocity ; [eqn: rm n ~=~ 2pi over □ c ~=~ m c]

In order to be stable, n must be real.

[eqn: rm therefore~~( n over m ) ^2 ~>~ 0]

Or      [eqn: rm{ □ m g_c } over { rho_l + rho _v } ~~>~~ {rho_l rho_v } over { ( rho_l + rho_v )^2 } ( V_v - V_l ) ^2]                              (1)

For steady-state, the continuity equations is

[eqn: - V_l ~=~ rho_v over rho_l V_v]                                            (2)

Inserting Eq.(2) into Eq.(1),        [eqn: rm {rho_l □ m g_c }over {rho_v ( rho_l + rho_v ) } ~~>~~ V_v ^2]

Thus, the maximum vapor velocity for a stable vapor stream will be

[eqn: rm V_v ~=~ left [ {rho_l □ m g_c }over {rho_v ( rho_l + rho_v ) } right ] ^1/2]                                     (3)

(2) Taylor instability.

□ > K.E & P.E                                                 (4)

→ Stable film boiling condition for a horizontal surface

① Total energy per wavelength of progressing wav due to K.E & P.E

: [eqn: rm left ( E over □ right )_{KE + PE} ~=~ { g ( rho_l - rho_v ) □^2 } over 2g_c]                                    (5)

where, [eqn: □] : max. amplitude of wave in Lamb's paper.

② Energy of wave due to surface tension

: [eqn: rm left ( E over □ right )_□ ~=~ 1 over □ int _o ^□ △ p ``□ ``dx]                                       (6)

The perssure differential for a curved surface of the wave form

[eqn: □ ~=~ □_o ``sin mx ~=~ □_o ``sin 2pi over □ x]

can be evaluated from a force balance on ds of the

surface

[eqn: rm △P ds ~=~ □ sin(d□) ~□~ □d□]

∴ [eqn: rm △ P ~=~ □ d□ over ds ~=~ □ 1 over R ~□~ □ {d^2 □ } over {d x^2 } ``=`` left ( 2pi over □ right )^2 □ □_o ``sin ``m``x]

∴[eqn: rm left ( E over □ right ) _□ ~=~ { □ □_o ^2 } over □ cdot 2pi over □ cdot pi]

∴ [eqn: rm □_crit ```<``` 2 pi`` root { {g_c ``□ } over { g ``(``rho_l ``-`` rho_v ``) }}]       for stable film boiling

Note that [eqn: rm □_crit ~∝~ "bubble diameter"]

and mininum unstable Talyor wavelength is [eqn: rm □_C = □_D over 3]