传质分离严格模拟计算

1、传质分离严格模拟计算第五章 传质别离过程的严格模拟计算 5.1 平衡级的理论模型 5.2 三对角矩阵法 5.3 同时校正法 5.4 多组分别离非平衡模型2多组分精馏过程简捷计算精馏简捷计算FUG法FenskeNmUnderwoodRmGillilandR、N多组分吸收过程简捷计算3严格计算的必要性简捷算法中引入的假设“恒摩尔流和相对挥发度为常数，在高压及塔顶、塔釜温差很大的情况下，偏向太大。 简捷计算不能给出各塔板上的浓度、温度等信息，也难以处理具有多股进料，多股侧采及有侧线换热等复杂别离过程。 特殊精馏，多组分吸收，多组分萃取等过程也以采用严格计算为宜。 4 Except for simpl

2、e cases, such as binary distillation, graphical, empirical, and approximate group methods are suitable only for preliminary design studies. Final design of multistage equipment for conducting multicomponent separations requires rigorous determination of temperatures, pressures, stream flow rates, st

3、ream compositions, and heat transfer rates at each stage. However, rigorous calculational procedures may not be justified when multicomponent physical properties or stage efficiencies are not reasonably well known.5严格计算的原那么：在给定的条件下，对每块塔板同时进展物料衡算，热量衡算及相平衡和归一化计算。 常用计算软件：Aspen Plus，HYSIM，Process II等。65

4、.1 平衡级的理论模型 UjWjFjLjVjLj-1Vj+1Qjzi,j, HF,j, TF,j, pF,jyi,j+1, Hj+1, Tj+1, pj+1xi,j, hj, Tj, pjxi,j-1, hj-1, Tj-1, pj-1yi,j, Hj, Tj, pj第j级平衡级General equilibrium stage7Assume that:(1) phase equilibrium is achieved at each stage 各级上到达相平衡(2) entrainment of liquid drops in vapor and occlusion of vapor bu

5、bbles in liquid are negligible 忽略雾沫夹带假定各级上到达相平衡且无化学反响。 8建模原那么：M Mass balance 物料衡算； MESH方程E Equilibrium relations 相平衡关系；S Summation equations 组分摩尔分率加和式； H Heat (energy, enthalpy) balance热量衡算。UjWjFjLjVjLj-1Vj+1Qjzi,j, HF,j, TF,j, pF,jyi,j+1, Hj+1, Tj+1, pj+1xi,j, hj, Tj, pjxi,j-1, hj-1, Tj-1, pj-1yi,j

6、, Hj, Tj, pj第j级9对第j级：1物料衡算Meq.： 01111=+-+-+=+-jijjjijjjijjijjijMjiyWVxULzFyVxLG,)()(i=1,2,CC个 2相平衡关系式Eeq.： 0=-=jijijiEjixKyG,i=1,2,CC个 3摩尔分率加和式Seq.： 2个 c001=-= .,1=ijiSYjyGi=1,2,C001=-= .,c1=ijiSXjxGi=1,2,C4热量衡算式Heq.： 01111=-+-+-+=+-jjjjjjjjFjjjjjHjQHWVhULHFHVhLG)()(,1个 共有2C3个方程 10将上述N个平衡级按逆流方式串联: U

7、jWjFjLjVjLj-1Vj+1QjjF1V1Q11FNLNQNN普通的N级逆流装置 11UjWjFjLjVjLj-1Vj+1QjjF1V1Q11FNLNQNN设计变量分析： 固定设计变量Nx： 进料变量数： NC+2压力等级数： N可调设计变量Na： NC+3串级单元数： 侧线采出数： 传热单元数： 分配器数： 12N2N03N-1总设计变量数Ni= Nx+Na = N(C+6)-1 12设计变量的规定：设计型：关键组分的回收率或浓度及相关参数平衡级数，进料位置等操作型：到达的别离程度回收率或浓度平衡级数，进料位置及相关参数13对操作型问题可以指定以下变量： 1、进料信息：Fj、zij、T

8、Fj、PFj NC+2个 2、各级压力：Pj N个 3、各级侧线采出：Uj、Wj 2N1个 4、各级换热：Qj N个 5、级数：N 1个未知量： N(C+6)-1 1、液相组成：xi,j NC个 2、气相组成：yi,j NC个 3、液相流率：Lj N个 4、气相流率：Vj N个 5、各级温度：Tj N个N2C+3个有唯一解 14 The above relations are nonlinear algebraic equations that interact strongly. Consequently, solution procedures are relatively difficu

9、lt and tedious. A solution method is required to be programmed for a computer.155.2 三对角线矩阵法Tridiagonal Matrix Algorithm Equation-tearing Procedures方程解离法又称配对收敛法 按方程类型分组的多级别离过程的计算方法。 适宜操作型计算。 5.2.1 方程的解离方法及求解 5.2.2 泡点法BP法5.2.3 流率加和法SR法165.2.1 方程的解离方法及求解Equation-tearing Procedures 一、方程的解离：MESHME：工作方程算组

10、成SH校验方程校核方程算温度、流率求解： 液相组成xi,j，汽相组成yi,j，温度Tj，流率Vj或Lj。泡点法BP：用S检验T，用H检验V。流率加合法SR：用S检验V，用H检验T。1701111=+-+-+=+-jijjjijjjijjijjijMjiyWVxULzFyVxLG,)()(i=1,2,CMeq.Eeq.0=-=jijijiEjixKyG,i=1,2,C将相平衡关系E-eq.代入物料衡算方程M-eq.： 011111=+-+-+-jijijjjijjjijjijijjijxKWVxULzFxKVxL,)()(为消去L，从第1级到第j级作总物料衡算： 11VWUFVLjmmmjj-+

11、= 1m=+)(将上式代入修正的M-eq.，整理可得： 18jjijjijjijDxCxBxA=+-11,19第1级无液相采出，第N级无汽相采出: 5-820二、三对角线矩阵的托马斯法 追赶法托马斯法求解： (1) 先假定Tj和Vj； (2) 计算相平衡常数Ki,j，得到线性化的ME方程； (3) 高斯消去法，将5-8转化为二对角矩阵方程； 解出xi,N，xi,N，xi,1。(4) 一般情况下，xi,j不会满足S-eq.和H-eq.，用S-eq.和H-eq.作为收敛的校验方程，算出新的Tj和Vj； (5) 以算出的新的Tj和Vj为迭代值，返回 (1)； 21配对收敛方法的特点： 将两个校验方程

12、S-eq.和H-eq.与两个迭代变量分别配对。 根据不同的配对方案形成两种不同的算法:泡点法(BP法)和流率加和法(SR法)。22 For separators where the feed(s) contains only components of similar volatility (narrow-boiling case), a bubble-point (BP) method is recommended. For a feed(s) containing components of widely different volatility (wide-boiling case) o

13、r solubility, a sum-rates (SR) method is suggested.235.2.2 泡点法BP法Bubble-point Method BP法适用于窄沸程混合物的别离计算，如一般的精馏过程。 在此情况下，各平衡级上的传质过程主要依赖于两相流体的局部汽化和局部冷凝，平衡级温度就是泡点温度或露点温度，它们主要取决于两相组成，所以用组分的摩尔分数加和式，即S-方程来检验平衡级温度Tj是否正确。 精馏系统内的热量传递主要由潜热的变化引起，由此也引起两相流率的变化，所以用热量衡算方程，即H-方程来检验流率Vj是否正确。24规定： 进料：Fj,zi,j,TFj,PFj 压

14、力：pj 侧采：Uj,Wj 热负荷：Qj除Q1和QN 级数：N 回流量：L1 气相馏出量：V1开 始设定Tj、Vj初值解三对角线矩阵方程，求xi,j归一化xi,j泡点计算，求新的Tj，Vj计算冷凝器和再沸器的热负荷Q1和QNH-eq.计算新的Vj ；计算Lj调整Tj和Vj结 束yesno规定设计变量是否满足迭代收敛准那么BP法计算框图25StartSpecify conditionsspecify： feed：Fj,zi,j,TFj,PFj pressure：pj side streams：Uj,Wj heat load：Qj No. of stages：N reflux：L1 vapor d

15、istillate：V1Initialize tear variables Tj, VjCompute x by Thomas methodNormalize xi,j for each stageCompute new Tj form BP eq and yCompute Q1 and QNCompute new Vj and LjIs 0.01N?ExityesnoconvergedNot convergedAdjust tear variablesAlgorithm for BP Method26一、迭代变量Tj、Vj初值的给出 1、Vj: 用指定回流比、馏出量、进料量、侧线采出量，按恒

16、摩尔流假设给出一组Vj的初值。 Establish an initial set of Vj based on the assumption of constant molar interstage flows using the specified reflux, distillate, feed, and side-stream flow rates.27一、迭代变量Tj、Vj初值的给出 2、Tj: 1塔顶：气相采出：液相采出：气、液相混合：露点温度泡点温度泡、露点之间的温度2塔釜：釜液泡点温度线性内插，得到中间各级温度初值。 28二、归一化 Normalization 由于求三对角矩阵方

17、程时没有考虑S-eq.的约束，必须对得到的xi,j归一化。 =Cjijixx=i1jix,三、泡点方程的计算 (实际就是S-eq.)： 001=-= .,c1=ijiSYjyG29四、Vj的计算 通过物料衡算和热量衡算得到二对角线矩阵方程： 先求V3，再依次求出V4VN。30五、迭代收敛的标准 或更简单的： p186【例5-1】泡点法模拟精馏别离轻烃混合物315.2.3 流率加和法Sum-Rates Method SR法适用于宽沸程混合物的别离过程， 如吸收、解吸、气提和萃取等过程的计算。 The chemical components present in most absorbers an

18、d strippers cover a relatively wide range of volatility 组分的挥发度相差大. Hence, the BP method of solving the MESH equations will fail because calculation of stage temperature by bubble-point determination is too sensitive to liquid-phase composition 通过泡点计算的级温度对液相组成的变化太敏感and the stage energy balance is muc

19、h more sensitive to stage temperatures than to interstage flow rates 热量平衡对级温度比对级间流率敏感的多. 用S-方程计算流率；用H-方程计算级温度。32规定： 进料：Fj,zi,j,TFj,PFj 压力：pj 侧采：Uj,Wj 热负荷：Qj除Q1和QN 级数：N 回流量：L1 气相馏出量：V1开 始设定Tj、Vj初值解三对角线矩阵方程，求xi,j归一化xi,j和yi,j求新的Tj(k+1)= Tj(k)+Tj(k) Tj(k)通过托马斯法求解一偏导数矩阵方程(5-18)S-eq.计算Lj ；物料衡算计算Vj调整Tj和Vj结

20、 束yesno规定设计变量是否满足迭代收敛准那么SR法计算框图33StartSpecify conditionsspecify： feed：Fj,zi,j,TFj,PFj pressure：pj side streams：Uj,Wj heat load：Qj No. of stages：NInitialize tear variables Tj, VjCompute x by Thomas methodCompute new Lj from sum-rates ralation and new VjNormalize xi,j and calculate corresponding yi,j

21、and normalize yi,jCompute new TjIs 0.01N?ExityesnoconvergedNot convergedAdjust tear variablesAlgorithm for SR Method34(I) Initialization of Tj and Vj Vj: 根据气相进料和侧采，按恒摩尔流假设给出一组Vj的初值。 Assume a set of Vj values based on the assumption of constant molar interstage flows, working up from the bottom of th

22、e absorber using specified vapor feeds and any vapor side-stream flows.35一、迭代变量Tj、Vj初值的给出 2、Tj: Generally, an adequate initial set of Tj values can be provided by computing or assuming both the bubble-point temperature of an estimated bottoms product and the dew-point temperature of an assumed vapor

23、 distillate product; or computing or assuming bubble-point temperature if distillage is liquid or a temperature in-between the dew-point and bubble-point temperatures if distillate is mixed, and then determining the other stage temperatures by assuming a linear variation of temperature with stage lo

24、cation.36(II) Calculate new values of Lj and Vj +=CkjkjLL=i11jix,)()(Lj(k) is calculated from Vj(k) by material balance： Vj(k1) is obtained by total material balance for stage jN ： () -+-=N=jmmmmNjjWUFLLV1 Values of xi,j obtained by Thomas algorithm are not normalized at this step but are utilized d

25、irectly to produce new values of Lj by applying the sum-rates equation.37(III) Normalize xi,j and yi,jNormalize xi,j: =Cjijixx=i1jix,Calculate yi,j by E-eq.:0=-=jijijiEjixKyG,Normalize yi,j: =Cjijiyy=i1jiy,38(IV) Calculate new Tj Since enthalpies are generally nonlinear in temperature, an iterative

26、solution procedure is required, such as the commonly used Newton-Raphson method： This matrix of partial derivatives is called the Jacobian correction matrix which can be solve by employing Thomas algorithm to get the set of corrections Tj(k).jkjjkjjkjjDTCTBTA)(1)()(1=D+D+D+-39(V) Convergence criteri

27、on Or simply：p190【例5-2】流率加和法模拟吸收塔40Summary1. Rigorous methods are readily available for computer-solution of equilibrium-based models for multicomponent, multistage absorption, stripping, distillation, and liquid-liquid extraction.2. The equilibrium-based model for a countercurrent-flow cascade prov

28、ides for multiple feeds, vapor side streams, liquid side streams, and intermediate heat exchangers. Thus, the model can handle almost any type of column configuration.41Summary3. The model equations include component material balances, total material balances, phase equilibria relations, and energy

29、balances.4. Some or all of the model equations can usually be grouped so as to obtain tridiagonal matrix equations, for which an efficient solution algorithm is available.42Summary5. Widely used methods for iteratively solving all of the model equations are the bubble-point (BP) method, the sum-rate

30、s (SR) method, the simultaneous correction (SC) method, and the inside-out method.6. The BP method is generally restricted to distillation problems involving narrow-boiling feed mixtures.7. The SR method is generally restricted to absorption and stripping problems involving wide-boiling feed mixture

31、s or in the Isothermal Sum-Rates (ISR) form to extraction problems.431893: Supplements- Development of Equilibrium-based Models The fundamental equations for the equilibrium-based models were first published by Sorel. The equations consisted of material balances around top and bottom sections of equ

32、ilibrium stages, including a total condenser and a reboiler, and corresponding energy balances that included provision for heat losses. Graphs of phase-equilibrium data were used instead of equations. Sorels model was not widely applied because of its complexity.441921: Supplements- Development of E

33、quilibrium-based Models Sorels model was adapted to graphical solution techniques for binary systems by Ponchon and Savarit who used an enthalpy-concentration diagram. 451925: Supplements- Development of Equilibrium-based Models A much simpler, but less rigorous, graphical technique was developed by

34、 McCabe and Thiele, who eliminated the energy balances by assuming a constant-molar-overflow. McCabe-Thiele graphical method is applied even today for binary distillation because the method gives valuable insight into changes in phase compositions from stage to stage. 461938: Supplements- Developmen

35、t of Equilibrium-based Models A notable achievement was made by Smoker for the distillation of a binary mixture by assuming not only constant molar overflow, but also constant relative volatility between the two components. 471930s1950s: Supplements- Development of Equilibrium-based Models Two itera

36、tive, numerical methods were developed for obtaining a general solution to Sorels model for the distillation of multicomponent mixtures. The Thiele-Geddes method requires specification of NT, the feed stage, R, and the distillate flow rate, with the resulting distribution of the components. The Lewi

37、s-Matheson method computes the NT and the location of the feed stage for a specified R and split between two key components. These two methods were widely used for the simulation and design of single-feed multicomponent distillation columns prior to the 1960s. 481958: Supplements- Development of Equ

38、ilibrium-based Models Techniques of matrix algebra were applied by Amundson, leading to a number of successful computer-aided design and simulation programs abound for the rigorous, iterative numerical solution of Sorels equilibrium-based model for a wide variety of column configurations and specifi

39、cations. Although the iterative computations sometimes fail to converge, the methods are widely applied and have become more flexible and robust with each passing year. 49Limitation of Equilibrium-based Models The equilibrium-based methods assume that equilibrium is achieved, at each stage, with res

40、pect to both heat and component mass transfer. Except when temperature changes significantly from stage to stage, the assumption of temperature equality for vapor and liquid phases leaving a stage is usually acceptable. However, in most industrial applications, the assumption of equilibrium with res

41、pect to exiting phase compositions is not reasonable. In general, exiting vapor-phase mole fractions are not related to exiting liquid-phase mole fractions by thermodynamic K-values. 50Procedures for accounting for nonequilibrium:Overall stage efficiency: Proposed by Lewis in 1922, For converting theoretical stages to actual stages. Experimental data show that this efficiency varies over a range of 5% to 120% dependi