双语化学辞典Chem Lexicon
研究生Graduate 环境化学与化学反应工程 Environmental chemistry & reaction engineering

理想反应器设计方程（CSTR 与 PFR）Ideal reactor design equations (CSTR and PFR)

\[\text{CSTR: } \tau = \frac{V}{Q} = \frac{c_{A0}X}{-r_A(X)},\qquad \text{PFR: } \tau = c_{A0}\int_0^X\frac{dX}{-r_A},\qquad \text{一级：}\ X_\text{CSTR} = \frac{k\tau}{1 + k\tau},\ X_\text{PFR} = 1 - e^{-k\tau}\]\[\text{CSTR: } \tau = \frac{V}{Q} = \frac{c_{A0}X}{-r_A(X)},\qquad \text{PFR: } \tau = c_{A0}\int_0^X\frac{dX}{-r_A},\qquad \text{first order :}\ X_\text{CSTR} = \frac{k\tau}{1 + k\tau},\ X_\text{PFR} = 1 - e^{-k\tau}\]
达姆科勒数（反应速率 / 输送速率）Damköhler number
\[\mathrm{Da} = k\tau c_{A0}^{\,n-1}\]

含义Meaning

连续搅拌釜（CSTR）内各处组成均匀、等于出口组成；平推流（PFR）内流体像活塞一样推进、无轴向混合。同一转化率下，正级数反应所需 PFR 体积小于 CSTR。

A CSTR is perfectly mixed at the exit composition; a PFR has plug flow with no axial mixing. For positive-order kinetics a PFR needs less volume than a CSTR for the same conversion.

符号说明Symbols

符号Symbol含义 Meaning单位 Unit
\(\tau\)空时（停留时间）space time (residence time)s
\(Q\)体积流量volumetric flow ratem³·s⁻¹
\(X\)转化率conversion1
\(-r_A\)A 的消耗速率rate of consumption of Amol·m⁻³·s⁻¹

推导Derivation

  1. CSTR 稳态物料衡算：\(F_{A0} - F_A - (-r_A)V = 0\)，反应速率按出口浓度计算。

    CSTR steady-state balance with the exit-concentration rate.

  2. PFR：对微元 dV 衡算 \(dF_A = r_A\,dV\)，积分。

    PFR: balance a slice dV and integrate.

  3. 一级反应 \(-r_A = kc_A\)：分别代入得两式。

    Insert first-order kinetics.

  4. 列文斯皮尔图：1/(−r_A) 对 X 作图，PFR 体积为曲线下面积，CSTR 为矩形面积。

    Levenspiel plot: PFR = area under 1/(−r) curve, CSTR = rectangle.

适用条件与局限Assumptions & limitations

  • 理想流型；真实反应器用停留时间分布（RTD）、多釜串联或轴向扩散模型描述。等温、恒密度假设。

    Ideal flow; real reactors need residence-time distributions. Isothermal, constant density.