Tab. 2-1 Aproksymacje członów równania ciągłości różnicami skończonymi
Człon | Aproksymacja schematem różnicowym |
$$\Delta Q$$ | $$(Q_{j+1}-Q_j)+\theta(\Delta Q_{j+1}-\Delta Q_j)$$ |
$$\frac{\partial A_c}{\partial t} \Delta x_c$$ | $$ 0.5\Delta x_{cj}\frac{\left( \frac{dA_c}{dz} \right)_j \Delta z_j + \left( \frac{dA_c}{dz} \right)_{j+1} \Delta z_{j+1}}{\Delta t}$$ |
$$ \frac{\partial A_f}{\partial t} \Delta x_f $$ | $$0.5\Delta x_{fj}\frac{\left( \frac{dA_f}{dz} \right)_j \Delta z_j + \left( \frac{dA_f}{dz} \right)_{j+1} \Delta z_{j+1}}{\Delta t}$$ |
$$ \frac{\partial S}{\partial t} \Delta x_f$$ | $$0.5\Delta x_{fj}\frac{\left( \frac{dS}{dz} \right)_j \Delta z_j + \left( \frac{dS}{dz} \right)_{j+1} \Delta z_{j+1}}{\Delta t}$$ |
Tab. 2-2 Aproksymacje członów równania pędu różnicami skończonymi
Człon | Aproksymacja |
$$\frac{\partial(Q_c \Delta x_c+Q_f\Delta x_f)}{\partial t \Delta x_e}$$ | $$\frac{0.5}{\Delta x_e \Delta t} (\Delta Q_{cj}\Delta x_{cj}+\Delta Q_{fj}\Delta x_{fj}+\Delta Q_{cj+1}\Delta x_{cj}+\Delta Q_{fj+1}\Delta x_{fj}) $$ |
$$\frac{\Delta \beta VQ}{\Delta x_{ej}}$$ | $$\frac{1}{\Delta x_{ej}} \left[ (\beta VQ)_{j+1}-(\beta VQ)_{j}\right] +\frac{\theta}{\Delta x_{ej}} \left[ \Delta(\beta VQ)_{j+1}-\Delta(\beta VQ)_{j}\right]$$ |
$$g\overline{A}\frac{\Delta z}{\Delta x_e}$$ | $$g\widetilde{A}\left[ \frac{z_{j+1}-z_j}{\Delta x_{ej}}+\frac{\theta}{\Delta x_{ej}}(\Delta z_{j+1}-\Delta z_j) \right] + \theta g \widetilde{A} \,\frac{z_{j+1}-z_j}{\Delta x_{ej}}$$ |
$$g\overline{A}(\overline{S}_f+\overline{S}_h)$$ | $$g\widetilde{A}(\widetilde{S}_f+\widetilde{S}_h)+ 0.5\theta\widetilde{A} \left[ (\Delta S_{fj+1}+\Delta S_{fj}) + (\Delta S_{hj+1}+\Delta S_{hj}) \right]+ 0.5g\theta (\widetilde{S}_f+\widetilde{S}_h)(\Delta A_j+\Delta A_{j+1})$$ |
$$\overline{A}$$ | $$0.5 (A_{j+1} + A_{j})$$ |
$$\overline{S}_f$$ | $$0.5(S_{f\,j+1}+S_{f\, j})$$ |
$$\partial A_j$$ | $$\left( \frac{dA}{dz} \right)_j \Delta z_j$$ |
$$\partial S_{f\,j}$$ | $$ \left( \frac{-2 S_f}{K}\frac{dK}{dz} \right)_j \Delta z_j + \left( \frac{2S_f}{Q} \right)_j \Delta Q_j $$ |
$$ \partial \overline{A} $$ | $$ 0.5 (\Delta A_{j+1} + \Delta A_j)$$ |