這一頁的公式我需要講解和實例---Third Chapter Theoretical Analysis
3-1-1 Theoretical Analysis of Direct Contact Membrane Distillation Mass Transfer Mechanism
In direct contact membrane distillation, because the saturated vapor pressures on both sides of the membrane are different, mass transfer occurs through the membrane. The permeate flux of water vapor through the membrane can be defined as:
N'' = c\_m (P\_1 - P\_2) (3.1)
Where N'' is the permeate flux; c\_m is the membrane permeability coefficient; P\_1 and P\_2 are the saturated vapor pressures of the feed side and permeate side membrane surface fluid respectively.
This study combines the theory of heat transfer analysis and converts the permeate flux into the following functional form, with the condition T\_1 - T\_2 < 15°C [16]:
N'' = c\_m |dP/dT|_{T\_m} (T\_1 - T\_2) = c\_m (P\_m λ M\_w) / (R T\_m²) (T\_1 - T\_2) (3.2)
Where T\_1 and T\_2 are the fluid temperatures on the feed side and permeate side membrane surface respectively; T\_m is the average temperature of the fluid on both sides of the membrane surface; λ is the latent heat of vaporization; M\_w is the molecular weight; R is the gas constant; P\_m is the average saturated vapor pressure of the fluid on both sides of the membrane surface. Because the feed streams in this study are pure water and 3.5 wt% salt water, when the salt water contains non-volatile solutes, it cannot be treated as an ideal solution. For non-ideal solutions, the activity coefficient a\_w [18] needs to be introduced to correct its deviation from ideal solution behavior.
Therefore, the relationship between the saturated vapor pressure of the feed side membrane surface fluid can be written as:
P\_1 = x\_w a\_w P\_w (3.3)
Where x\_w is the mole fraction of pure water in the salt water solution; P\_w is the saturated vapor pressure of pure water; a\_w is the activity coefficient. The activity coefficient of pure water in salt water solution can be calculated using the empirical formula [18]:
a\_w = 1 - 0.5x\_NaCl - 10x\_NaCl² , x\_NaCl < 0.097 (3.4)
19
第三章 理論分析
其中，$x_{NaCl}$ 為氯化鈉於鹽水溶液中的莫耳分率。經由式(3.3)與式(3.4)，遂
將兩端薄膜表面流體之平均飽和蒸氣壓整理為：
$P_m = ((1-x_{NaCl})(1-0.5x_{NaCl}-10x_{NaCl}^2)P_w + P_2)/2$ (3.5)
將式(3.5)帶入式(3.2)，則可得透膜通量之溫度關係式如下：
$N'' = \left[c_m \frac{((1-x_{NaCl})(1-0.5x_{NaCl}-10x_{NaCl}^2)P_w + P_2)\lambda M_w}{2RT_m^2}\right](T_1-T_2)$ (3.6)
將(3.6)式完全轉換為溫度函數，此處引入安東尼方程式(Antoine equation)，
用以表示純水飽和蒸氣壓與溫度的關係：
$P = \exp\left(23.238 - \frac{3841}{T-45}\right)$ (3.7)
在薄膜蒸餾系統中，薄膜係數 $c_m$ 對於計算透膜通量扮演相當重要的
角色，其表示用以描述水蒸氣分子在薄膜孔洞中的運動模式，有助於準確
預測通過薄膜之氣體分子的多寡。而氣體分子通過疏水性薄膜時，可以由
下列模式描述：(1)當薄膜孔徑小於蒸氣分子的平均自由路徑(mean free path)，
蒸氣分子與薄膜孔壁的碰撞頻率大於蒸氣分子自身碰撞頻率，此時可用克
努森擴散(Knudsen diffusion)模式來表示；(2)而蒸氣分子的平均自由路徑
遠小於薄膜孔徑時，蒸氣自身碰撞頻率則遠大於與薄膜孔壁的碰撞頻率，
此時蒸氣之流動方式適用泊醇流動(Poiseuille flow)模式來描述；(3)倘若存
在於薄膜孔洞中之空氣為不凝結氣體，因此將在薄膜孔洞中形成一種阻力
而阻礙蒸氣分子的流動，此現象適用傳統之分子擴散(Molecular diffusion)