diff --git a/docs/src/P3Scheme.md b/docs/src/P3Scheme.md index 0c8ffbbe25..c53cbcb147 100644 --- a/docs/src/P3Scheme.md +++ b/docs/src/P3Scheme.md @@ -125,10 +125,10 @@ As a result ``q\_{ice}`` can be expressed as a sum of inclomplete gamma function | condition(s) | ``q_{ice} = \int \! m(D) N'(D) \mathrm{d}D`` | gamma representation | |:---------------------------------------------|:-----------------------------------------------------------------------------------------|:---------------------------------------------| | ``D < D_{th}`` | ``\int_{0}^{D_{th}} \! \frac{\pi}{6} \rho_i \ D^3 N'(D) \mathrm{d}D`` | ``\frac{\pi}{6} \rho_i N_0 \lambda \,^{-(\mu \, + 4)} (\Gamma \,(\mu \, + 4) - \Gamma \,(\mu \, + 4, \lambda \,D_{th}))``| -| ``q_{rim} = 0`` and ``D > D_{th}`` | ``\int_{D_{th}}^{\infty} \! \alpha_{va} \ D^{\beta_{va}} N'(D) \mathrm{d}D`` | ``\alpha_{va} \ N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1) + \Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{th}) - (\mu \, + \beta_{va} \,)\Gamma \,(\mu \, + \beta_{va} \,))`` | +| ``q_{rim} = 0`` and ``D > D_{th}`` | ``\int_{D_{th}}^{\infty} \! \alpha_{va} \ D^{\beta_{va}} N'(D) \mathrm{d}D`` | ``\alpha_{va} \ N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{th}))`` | | ``q_{rim} > 0`` and ``D_{gr} > D > D_{th}`` | ``\int_{D_{th}}^{D_{gr}} \! \alpha_{va} \ D^{\beta_{va}} N'(D) \mathrm{d}D`` | ``\alpha_{va} \ N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{th}) - \Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{gr}))`` | | ``q_{rim} > 0`` and ``D_{cr} > D > D_{gr}`` | ``\int_{D_{gr}}^{D_{cr}} \! \frac{\pi}{6} \rho_g \ D^3 N'(D) \mathrm{d}D`` | ``\frac{\pi}{6} \rho_g N_0 \lambda \,^{-(\mu \, + 4)} (\Gamma \,(\mu \, + 4, \lambda \,D_{gr}) - \Gamma \,(\mu \, + 4, \lambda \,D_{cr}))`` | -| ``q_{rim} > 0`` and ``D > D_{cr}`` | ``\int_{D_{cr}}^{\infty} \! \frac{\alpha_{va}}{1-F_r} D^{\beta_{va}} N'(D) \mathrm{d}D`` | ``\frac{\alpha_{va}}{1-F_r} N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1) + \Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{cr}) - (\mu \, + \beta_{va} \,)\Gamma \,(\mu \, + \beta_{va} \,))`` | +| ``q_{rim} > 0`` and ``D > D_{cr}`` | ``\int_{D_{cr}}^{\infty} \! \frac{\alpha_{va}}{1-F_r} D^{\beta_{va}} N'(D) \mathrm{d}D`` | ``\frac{\alpha_{va}}{1-F_r} N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{cr}))`` | where ``\Gamma \,(a, z) = \int_{z}^{\infty} \! t^{a - 1} e^{-t} \mathrm{d}D`` and ``\Gamma \,(a) = \Gamma \,(a, 0)`` for simplicity.