Food Process Design I (1503-420) Part I – Mass Transfer PDF

Food Process Design I (1503-420) Part I – Mass transfer R. Kohlus, P. Gschwind, M. Spiess, H. Teichmann, Y. Rudoph Institute of Food Science and Biotechnology Dept. Process Engineering and Food Powders University of Hohenheim – WS 2024/25...

Food Process Design I (1503-420) Part I – Mass transfer R. Kohlus, P. Gschwind, M. Spiess, H. Teichmann, Y. Rudoph Institute of Food Science and Biotechnology Dept. Process Engineering and Food Powders University of Hohenheim – WS 2024/25 1 Set-up of Master lectures in FPE Targets of the master training: „Engineering approach“ for food technologists How to tackle engineering problems. Which assumptions make sense What is the scientific basis of the topics Where do I find or get the needed information How to generate new knowledge (R&D as job target) 2  Lectures: Food science and Engineering FG LVT: Process Engineering and Food Powders FPD I: Transport Phenomena (O) FPD II: Line design & Scale up (E) PDPD: Process Product Interactions (E) Cereals & Sweets) Drying and Instantisation of Food (E) Encapsulation of Biofunctional Ingredients (E) (lead Food Biophysics) 3 Expectations - objectives What are your expectations for this course? - What do you want to get out? - Which question would you like to have answered? - What capabilities do you intend to acquire? - Any questions you always wanted to asks, but… - How does the lecture fit in the master context? 4 Food Process Design I Efficient Processing and Transport Phenomena Optimal process design: Effect /Effort = Efficiency Energy efficiency Investments Quality, Yield,… Transport phenomena Heat transport Mass transport Momentum transport 5 Efficiency not effectiveness „Efficiency“ quantifies the effect relative to the invested effort or resources. It is defined differently in different applications. E.G.: − Ratio of result and effort; relates to degree of efficiency − Ratio between data transfer rate and band width of communication (spectral eff.) − Product amount relative to energy requirements (energy efficiency) − Product amount relative to environmental effects (ecological efficiency) An optimum is defined by conditions resulting in highest efficiency, that is given by maximum results at given resources or defined result with minimum effort. One of the two not both at the same time. It can only be defined with one specified target parameter. For details see FPD II. 6 Modeling in engineering Mass and Heat Transfer: Analysis of Mass Contactors and Heat Exchangers, T. W. FRASER RUSSELL, ANNE S. ROBINSON, NORMAN J. WAGNER , Univ. Delaware All physical situations of interest to engineers and scientists are complex enough, that a mathematical model of some sort is essential to describe them in sufficient detail for useful analysis and interpretation. Mathematical expressions provide a common language so different disciplines can communicate among each other more effectively. Models are very critical to chemical engineers, chemists, biochemists, and other chemical professionals because most situations of interest are molecular in nature and take place in equipment that does not allow for direct observation. Experiments are needed to extract fundamental knowledge and to obtain critical information for the design and operation of equipment. To do this effectively, one must be able to quantitatively analyze mass, energy, and momentum transfer (trans- port phenomena) at some level of complexity. In this text we define six levels of complexity, which characterize the level of detail needed in model development. 7 Course topics 2 Transport phenomena Applications: thermal separation technologies Destilling, Rectification Concentrating Absorbtion Purifying Adsorbtion removal production Extraction protection Drying Release Crystalisation Additon / doting Encapsulation Unit Operations – Mass Transfer Unit operation which are driven by mass transfer – Phase equilibria processes Adsorption/Desorption Gas-Solid or Liquid-Solid Sublimation/Desublimation Gas-Solid Absorption/Desorption Liquid-Gas Distillation Liquid-Gas Evaporation/ condensation Liquid-Gas Extraction Liquid-Liquid or Liquid-Solid Drying Liquid-Solid Crystallisation Liquid-Solid Content 1 Introduction 5 Thermal Separation Technics a.     Efficiency/ Effectivity a.     Destillation / Rectification b.    Applications b.    Absorption c.    Analogy Mass & Heat transfer c.     Adsorption d.    Extraction 2 Masstranfer 6 Drying a.     Diffusion coefficients a.    Coupled Heat and Mass Transfer b.    Fick vs. Maxwell Stefan b.    Sorption c.    One sided Diffusion c.     Normalised drying curves d.    Multi component Diffusion d.    Spezial Drying Technics 3 Heat Transfer 5.1.1 Phasequilibria and activity coefficients a.    Transfer mechanism a.     Van Laar; NRTL; Wilson; Uniquac; Unifac; CRS b.   Calculaction and Design of HE b.    Solid –liquid behaviour - Glasses c.    Non newtonian Media 8 Membrane separation 4 Fluid mechanics 9 Crystallisation a.    Balance Equations 10 Chemical Reaction engineering b.   Turbulence a.     Residence time distrubtions: CSTR; IPR c. Boundary layer b.    Order of reactions c.     Mass transfer with chem. Reactions 10  1.1 Literature Basic Literatur – Of general importance for the lecture 1 Mersman, Stichlmair, Kind Thermal Separation Technology Springer Verlag, Berlin (2011) Verlag Oldenbourg München, Wien 2 Lohrengel, B. Einführung in die thermischen Trennverfahren (2007) 3 Sattler, K. Thermal Separation Processes. Principles and Design VCH-Verlag Weinheim (2007) Transportvorgänge in der Verfahrenstechnik 4 Kraume, M. Grundlagen und apparative Umsetzungen Springer-Verlag (2004) 5 Vauck-Müller Grundoperationen chem. Verfahrenstechnik VCH-Verlag Weinheim (1988) 6 Billet Industrielle Destillation Verlag Chemie, Weinheim (1973) 7 Perry Chemical Engineers´ Handbook 8 VDI-Heat Atlas Springer Verlag, 10.Auflage (2010) 9 Hartel, R. W. Crystallization in Foods Aspen Publishers, Inc. (2001) 10 Mersmann, A. Crystallization Technology Handbook Marcel Dekker, Inc. (2001) 11 Atkins, P. W. Physical chemistry WILEY-VCH Verlag GmbH (2001) 12 Kessel, H. G. Food and Bio Process Engineering Druckerei Rieder GmbH (2002) 13 Levenspiel, O. Chemical Reaction Engineering John Wiley & Sons Ltd. (1962, 1972) 14 Bird, Steward, Lightfood Transport Phenomena John Wiley and Sons Transport phenomena - Causes Heat transport Mass transport Heat conduction Mass conduction (Diffusion) Convective heat transport Convective mass transport Radiation - …. Depending on: Depending on: -Mechanism - Mechanism - Driving Force (Treibendes Gefälle) - Driving Force (Treibendes Gefälle) - Temperature -concentration difference - Interface -Interface 12 Applications Drying of a solid body T = drying body L = Air stream + = H2O-vapour Wet scrubber – Air conditioning S = Nozzle L = air flow T = drop separator o = H2O-drops + = H2O-vapour Applications Separation of CO2 from a gas by absorption and reaction Dissolution of particles Recapitulation: Stationary transfer Input c1 Output c2 c1 c2 Conzentration c Lineary dependency of c1 concentration c and diffusion distance x c2 Distance x Mass transport by Diffusion Diffusion: (molecular) Mass transport by microscopic relative movement. (Massetransport durch mikroskopische Relativbewegung von Teilchen. /Brockhaus Physik/) 1. Fick‘ian Law List of Symbols Ṅ i,x in x-direction, perpendicular to plane A mol diffusing molecular flux of type i 𝜕𝜕𝑐𝑐𝑖𝑖 s 𝑁𝑁̇ 𝑖𝑖,𝑥𝑥 = −𝐷𝐷𝑖𝑖 𝐴𝐴 A Diffusion plane m2 𝜕𝜕𝜕𝜕 𝜕𝜕ci Concentration gradient in direction of mol 𝜕𝜕𝜕 diffusion flux m3 m x Coordinate of diffusion space m Di Diffusivity of component i m2 s Food Process Design I - WS2024/25 17  Definitions: Driving forceS and transport phenomena 1. Fick‘ian Law Newton‘s shear Fourier‘s Law stress approach Mass transport Momentum transp. Heat transport 𝜕𝜕𝜕𝜕 𝜕𝜕𝑐𝑐𝑖𝑖 F= −𝜂𝜂 ⋅ 𝐴𝐴 𝜕𝜕𝑇𝑇 𝑁𝑁̇ 𝑖𝑖,𝑥𝑥 = −𝐷𝐷𝑖𝑖 ⋅ 𝐴𝐴 ⋅ 𝜕𝜕𝜕𝜕 𝑄𝑄̇ 𝑥𝑥 = −𝜆𝜆 ⋅ 𝐴𝐴 ⋅ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑐𝑐𝑖𝑖 𝜕𝜕𝜕𝜕 𝜏𝜏 = −𝜂𝜂 ⋅ 𝜕𝜕𝑇𝑇 𝑛𝑛̇ 𝑖𝑖,𝑥𝑥 = −𝐷𝐷𝑖𝑖 ⋅ 𝜕𝜕𝜕𝜕 𝑞𝑞̇ 𝑥𝑥 = −𝜆𝜆 ⋅ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 List of Symbols List of Symbols List of Symbols Ṅ i,x in x-direction, perpendicular to plane A mol ẋ x flow in x-direction, perpendicular to m q̇ ,x Heat flux in x-direction, perpendicular J diffusing molecular flux of type i = 𝑢𝑢 plane A s to plane A = 𝑊𝑊 s s A Diffusion plane m2 A flow crosss section m2 A Heat transfer plane m2 𝜕𝜕ci Concentration gradient in direction of mol 𝜕𝜕𝑢𝑢 Velocity gradient perpendicular to flow 1 𝜕𝜕𝑇𝑇 Temperature gradient in direction of 𝐾𝐾 𝜕𝜕𝜕 diffusion flux m3 m 𝜕𝜕𝜕 direction: Shear rate s 𝜕𝜕𝜕 heat flux m x Coordinate of diffusion space m x space coordinate of flow m x Coordinate of heat conduction m Di Diffusivity of component i 2 dynamic viscosity 2 𝜆𝜆 Heat conductivity 𝑊𝑊 m 𝜂𝜂 m s s m2 s Gradient - Divergence Three dimensional notation Carthesian Cylinder coordinate Spherical coordinates Gradient - Divergence Sources and sinks are determined by the divergence ▼ is called NABLA operator Derivation - 2. Fick‘ian law/ 2. Fourier law 𝜕𝜕𝑐𝑐 𝜕𝜕𝜕𝜕 𝜕𝜕𝑛𝑛̇ 𝑖𝑖 𝜕𝜕 −𝐷𝐷 𝑖𝑖 𝜕𝜕 2 𝑐𝑐𝑖𝑖 𝜕𝜕𝜕𝜕 =− =− = 𝐷𝐷 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑥𝑥 2 D constant Derivation by combining1. Fickian law and Continuity equation Derivation – Fourier Kirchhoff law 𝜕𝜕𝑐𝑐 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑛𝑛̇ 𝑖𝑖 𝜕𝜕 −𝐷𝐷 𝑖𝑖 𝜕𝜕 2 𝑐𝑐𝑖𝑖 𝜕𝜕𝜕𝜕 =− =− = 𝐷𝐷 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑥𝑥 2 D constant Derivation by combining 1. Fick‘ian law and Continuity equation Derivation - 2. Fick‘ian law/ 2. Fourier law 𝜕𝜕𝑐𝑐𝑖𝑖 𝜕𝜕𝜕𝜕 𝜕𝜕𝑛𝑛̇ 𝑖𝑖 𝜕𝜕 −𝐷𝐷 𝜕𝜕 2 𝑐𝑐𝑖𝑖 𝜕𝜕𝜕𝜕 =− =− = 𝐷𝐷 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑥𝑥 D Diffusivity constant Derivation by combining1. Fickian law and Continuity equation Instationary heat conduction Differential equation (2. FOURIER‘s law) One dimensional, instationary heat conduction, if λ constant with x: 𝜕𝜕𝜕𝜕 𝜆𝜆 𝜕𝜕 2 𝑇𝑇 𝜕𝜕 2 𝑇𝑇 = ⋅ = 𝑎𝑎 ⋅ 2 𝜕𝜕𝜕𝜕 𝑐𝑐𝑝𝑝 ⋅ 𝜌𝜌 𝜕𝜕𝑥𝑥 2 𝜕𝜕𝑥𝑥 𝑇𝑇 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 [𝐾𝐾] 𝑡𝑡 = 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍 [𝑠𝑠] 𝑐𝑐𝑝𝑝 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 [𝑘𝑘𝑘𝑘/𝑘𝑘𝑘𝑘 𝐾𝐾] 𝜌𝜌 = 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 [𝑘𝑘𝑘𝑘/𝑚𝑚𝑚] three dimensional, instationary heat cond. 𝜆𝜆 = 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 [𝑊𝑊/𝑚𝑚 𝐾𝐾] 𝑎𝑎 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝜆𝜆/ 𝑐𝑐𝑝𝑝 ⋅ 𝜌𝜌 [𝑚𝑚𝑚/𝑠𝑠] 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑖𝑖𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑇𝑇 𝜕𝜕 2 𝑇𝑇 𝜕𝜕 2 𝑇𝑇 = 𝑎𝑎 ⋅ + + 𝜕𝜕𝜕𝜕 𝜕𝜕𝑥𝑥 2 𝜕𝜕𝑦𝑦 2 𝜕𝜕𝑧𝑧 2 The solutions of the PDE depends on the starting and boundary conditions of each problem. Only simple case can be solved analytically: e.g. temperature profiles in plates, cylinders and spheres. Instantionary Heat transfer if 37 Descriped by Fourier‘sche PDE. or 2. Fick‘s Law respectively Cartesian coordinates (x,y,z) Cylinder coordinates (r,ϕ,z) Cylinder coordinates: x = r cos ϕ; y = r sin ϕ; z = z Spherical coordinates (ϕ,ψ,z) Spherical coordinates: x = r sin ψ cos ϕ; y = r sin ψ sin ϕ; z = r cos ϕ;  Cylinder coordinates: One dimensional problems (relative to r; i.e. x=r) x = r cos φ; y = r sin φ; z = z For systems with symmetry the Fourier PDE Sphere coordinates: Can be written in similar form, with parameter b. x = r sin ψ cos φ; y = r sin ψ sin φ; z = r cos φ; 1 for plates b= 2 for cylinder 3 for spheres 39 Heat conduction / Diffusion Dimensionless denotation 𝑥𝑥 𝑥𝑥 𝜂𝜂 = 𝑏𝑏𝑏𝑏𝑏𝑏. 𝜂𝜂 = 2 𝑎𝑎𝑎𝑎 2 𝐷𝐷𝐷𝐷 Nach: Grundlagen der Wärme- und Stoffübertragung R. Jeschar, W. Alt, E. Specht Heat conduction / Diffusion - Boundary conditions Boundary conditions: The temperature development in a body are determined by ist thermodynamic initial state and its interaction with the environment. As the Fourier-Kirchhoff equation is a PDE of 1. order by time and 2. order by space we need at least 1 intial (time) and 2 boundary conditions (space). In n-dimensional conduction problems (n = 1,2,3 ) we need 2 × n boundary conditions. a) Initial conditions The temperature function (field) T(x,y,z,t) at time t = 0, is given. That is the starting situation with respect to temperature, the change of which will be calculated. It is the tempearture distribution for all points of the space of interest: Tt=0(x,y,z). In many case this initial temperature will be taken as constant Tt =0 within the body: Heat conduction / Diffusion - Boundary conditions Boundary conditions: b) Boundary conditions:(short: „BC“) For PDE of 2. order typically BC (spatial BC)are classified in three types. BC of 1. type is given, when a function is given for the boundary. If the gradient at the boundary is given ist termed 2. type, a combination 3. type. In the case of heat conduction: 1. Type: (Dirichlet‘s BC): At the boundary of the body the temperature is given as a function of time: „For a border point „T = Tw“ Index w: wall. For example, this is the case when the surface temperature is given by measurements (i.e. by a pyrometer or Infrared thermografy) or in the case of phase transitions (the surface is at melting or condensation temperature), or in case of ideal contact of two bodies.. Heat conduction / Diffusion - Boundary conditions Boundary conditions : 2. type: (Neumann‘s BC): At the boundary (border) the temperature gradient is known (given). Another option is that the heat flux density is given, which is directly related to the gradient. The BC is given when the heat flux at surface is fixed, i.e. in case of electrical heating. 3. Type: (Cauchy‘s BC): The environmental conditions (Index u) and the heat transfer mechanism between boundary and environment are known: The temperature gradient at the body surface (boundary) is, (different from type 2) not a given independent function, but varies with surface temperature of the body. Nach: Grundlagen der Wärme- und Stoffübertragung R. Jeschar, W. Alt, E. Specht Instationary Heat transfer 44 Heat transfer / Comsol 45 Temperature profile in permeation zone Ingress of heat into infinite extended bodies (instationary, one dim.) Boundary conditions: Instationary heat flux in solid bodies with huge dimensions (1/Bi t1 ∞ t1 > t0 x For simple geometries (Plate, Cylinder, sphere) solutions to the PDE are known (see Heat transfer) Analogy of Mass and heat transfer Analogy of Mass and heat transfer General: Analogy is given when PDE of different physical process are formally equal (also same starting and boundary conditions) Diffusion in non-ideal systems Interactions between „Micro“- and Macromolekules ⇒ Deviations from ideal behaviour, diffusion not predictable without experiments (effective Diffusion coefifcient Deff) Problems in prediction and calculation of mass transfer and diffusion in food processing: Why? Diffusion in non-ideal systems Interactions between „Micro“- and Macromolekules ⇒ Deviations from ideal behaviour, diffusion not predictable without experiments (effective Diffusion coefifcient Deff Problemes in prediction and calculation of mass transfer and diffusion in food processing: Why? - Complex structures (Multiphases, gels, …) - Intermoleculare interactions - Variations in chemical composition - Chemical reactions in parallel ⇒ Systems can change during storage and processing ⇒ Non ideal condition=> Instationary processes Conditions of mass tranfer Mass transfer can happen in different conditions => resulting in different description approaches E.g. ⇒ Molecular Diffusion ⇒ Convective Mass transport ⇒ Molecule assemblies move (bulk flow):  Free / natural flow (free convection)  Forced flow (forced convection) Mass transport by convection 𝑛𝑛𝑖𝑖 Ph ci,k ci ci,G 𝛿𝛿 Phase I Phase II x Ph = Phase boundary ci = concentration of transported substance x = spatial coordinate ni = molecular flow of transported substance δ = Boundary layer ci,k = Concentration in center of phase I ci,G = Concentration at phase boundary Laminar flow in boundary layer Assumption: Laminar flow in boundary layer : => Mass transport happens only by molecular diffusion Equation for βi: 1. Fick‘ian law: Δ𝑐𝑐𝑖𝑖 𝑛𝑛𝑖𝑖 = −𝐷𝐷𝑖𝑖 ⋅ 𝛿𝛿 Δ𝑐𝑐𝑖𝑖 𝐷𝐷𝑖𝑖 𝑛𝑛𝑖𝑖 ⋅ 𝐴𝐴 = −𝐷𝐷𝑖𝑖 ⋅ 𝐴𝐴 ⋅ = 𝛽𝛽𝑖𝑖 ⋅ 𝐴𝐴 ⋅ Δ𝑐𝑐𝑖𝑖 => 𝛽𝛽𝑖𝑖 = − 𝛿𝛿 𝛿𝛿 𝑛𝑛𝑖𝑖 =: 𝛽𝛽𝑖𝑖 ⋅ Δ𝑐𝑐𝑖𝑖 Mass transport equations Approximately we can write 𝛽𝛽𝑖𝑖 ≈ −𝐷𝐷𝑖𝑖 /𝛿𝛿 1 = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 tan 𝑐𝑐 𝑒𝑒 𝛽𝛽𝑖𝑖 (only in case of laminar boundary layer Dependent on and molecular diffusion) − Geometry − Material properties − Flow conditions Laminar flow in boundary layer Assumption: Laminar flow in boundary layer : => Mass transport happens only by molecular diffusion Equation for βi: 1. Fick‘ian law: 𝜕𝜕𝑐𝑐𝑖𝑖 Δ𝑐𝑐𝑖𝑖 𝑛𝑛𝑖𝑖 = −𝐷𝐷𝑖𝑖 ⋅ = −𝐷𝐷𝑖𝑖 ⋅ 𝜕𝜕𝜕𝜕 𝛿𝛿 Δ𝑐𝑐𝑖𝑖 𝐷𝐷𝑖𝑖 𝑛𝑛𝑖𝑖 ⋅ 𝐴𝐴 = −𝐷𝐷𝑖𝑖 ⋅ 𝐴𝐴 ⋅ = 𝛽𝛽𝑖𝑖 ⋅ 𝐴𝐴 ⋅ Δ𝑐𝑐𝑖𝑖 => 𝛽𝛽𝑖𝑖 = 𝛿𝛿 𝛿𝛿 𝑛𝑛𝑖𝑖 =: −𝛽𝛽𝑖𝑖 ⋅ Δ𝑐𝑐𝑖𝑖 Mass transport equations 𝛿𝛿 𝛿𝛿 1 𝜕𝜕𝑐𝑐 −𝐷𝐷𝑖𝑖 𝜕𝜕𝑐𝑐 𝛽𝛽𝑖𝑖 = −𝐷𝐷𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 ∆𝑐𝑐 𝜕𝜕𝑥𝑥 ∆𝑐𝑐 𝜕𝜕𝑥𝑥 0 0 Convective Mass transport in stationary conditions Gases 𝑁𝑁𝑖𝑖 = 𝛽𝛽𝑖𝑖 ∗ 𝐴𝐴 ∗ 𝑐𝑐𝑖𝑖,𝑘𝑘 − 𝑐𝑐𝑖𝑖,𝐺𝐺 = 𝛽𝛽𝑖𝑖 ∗ 𝐴𝐴 ∗ Δ𝑐𝑐𝑖𝑖 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑔𝑔𝑔𝑔𝑔𝑔 𝑙𝑙𝑙𝑙𝑙𝑙: 𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝛽𝛽𝑖𝑖 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 [𝑚𝑚/𝑠𝑠] 𝑝𝑝 ∗ 𝑉𝑉 = 𝑁𝑁 ∗ 𝑅𝑅 ∗ 𝑇𝑇 => 𝑓𝑓𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖: 𝑝𝑝𝑖𝑖 ⋅ 𝑉𝑉 = 𝑁𝑁𝑖𝑖 ⋅ 𝑅𝑅 ⋅ 𝑇𝑇 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐: 𝑁𝑁𝑖𝑖 𝑐𝑐𝑖𝑖 = 𝑉𝑉 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐿𝐿𝐿𝐿𝐿𝐿 𝑝𝑝𝑖𝑖 𝑝𝑝𝑖𝑖 𝑝𝑝𝑖𝑖 = 𝑐𝑐𝑖𝑖 ∗ 𝑝𝑝 => 𝑝𝑝 = 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜: 𝑐𝑐𝑖𝑖 = 𝑐𝑐𝑖𝑖 𝑝𝑝 𝑁𝑁𝑖𝑖 𝑝𝑝𝑖𝑖 Δ𝑝𝑝𝑖𝑖 => 𝑐𝑐𝑖𝑖 = = => Δ𝑐𝑐𝑖𝑖 = 𝑉𝑉 𝑅𝑅 ⋅ 𝑇𝑇 𝑅𝑅 ⋅ 𝑇𝑇 𝛽𝛽𝑖𝑖 ⋅ 𝐴𝐴 𝛽𝛽𝑖𝑖 ⋅ 𝐴𝐴 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑝𝑝𝑖𝑖,𝑘𝑘 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 pressure of moving gas in center 𝑁𝑁𝑖𝑖 = ⋅ 𝑝𝑝𝑖𝑖,𝑘𝑘 − 𝑝𝑝𝑖𝑖,𝐺𝐺 = Δ𝑝𝑝 𝑅𝑅 ⋅ 𝑇𝑇 𝑅𝑅 ⋅ 𝑇𝑇 𝑖𝑖 𝑝𝑝𝑖𝑖,𝐺𝐺 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 pressure of moving gas at interphase Prediction of mass tranfer coefficient SCHMIDT‘sche Number Sc: ν = kinematic Viskosity [m²/s] 𝜈𝜈 η = dynamic Viskosity [Pas] 𝑆𝑆𝑆𝑆 = 𝐷𝐷 ρ = Density [kg/m³] l = characteristic length [m] SHERWOOD Number Sh: w = flow speed [m/s] a1,a2,a3 = approximation parameter 𝛽𝛽 ⋅ 𝑙𝑙 𝑆𝑆𝑆 = 𝐷𝐷 (experimentaly determined) REYNOLDS-Number Re: 𝑤𝑤 ⋅ 𝑑𝑑 𝑤𝑤 ⋅ 𝑑𝑑 ⋅ 𝜌𝜌 Re = = 𝜈𝜈 𝜂𝜂 Mass transport by convection Sh = Mass transport by Diffusion Standard approach: 𝑆𝑆𝑆 = 𝑎𝑎1 ⋅ Re𝑎𝑎2 ⋅ 𝑆𝑆𝑐𝑐 𝑎𝑎3 Analogue Property Heat transport Masstransport Exchange property Heat flow Molecular flow 𝐽𝐽 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 𝑄𝑄 = 𝑊𝑊 𝑛𝑛 𝑠𝑠 𝑠𝑠 Mass flow 𝑘𝑘𝑘𝑘 𝑚𝑚 𝑠𝑠 Driving force Temperature gradient Concentration gradient Δ𝑇𝑇 𝐾𝐾 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 Δ𝑐𝑐𝑖𝑖 𝑚𝑚𝑚 Partial pressure Δ𝑝𝑝𝑖𝑖 𝑃𝑃𝑃𝑃 = 10−5 𝑏𝑏𝑏𝑏𝑏𝑏 𝑘𝑘𝑘𝑘 Δ𝜌𝜌𝑖𝑖 Partial density gradient´´ 𝑚𝑚𝑚 Transport coefficient Heat conductivity Diffusion coefficient 𝑊𝑊 𝑚𝑚𝑚 𝜆𝜆 𝐷𝐷 𝑚𝑚 ∗ 𝐾𝐾 𝑠𝑠 Analogy Analogue quantity Heat transport Mass transport Equations for transport a) Stationary 1. Fourier-Law 1. Fick‘sches Gesetz Δ𝑇𝑇 Δ𝑐𝑐 𝑞𝑞 = −𝜆𝜆 𝑛𝑛 = −𝐷𝐷 Δ𝑥𝑥 Δ𝑥𝑥 b) Instationary Fourier-Kirchhoff Law 2. Fick‘ian law 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕 = 𝑎𝑎 ∗ = 𝐷𝐷 ∗ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕 Transfer coefficient heat transfer coef. mass transfer coefficient 𝑊𝑊 𝑚𝑚 𝛼𝛼 𝛽𝛽 Equations for con- 𝑚𝑚𝑚 ∗ 𝐾𝐾 𝑠𝑠 vective transport 𝑞𝑞 = 𝛼𝛼 Δ𝑇𝑇 𝑛𝑛 = 𝛽𝛽 Δ𝑐𝑐 𝑞𝑞 = 𝑘𝑘𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 ∗ Δ𝑇𝑇 𝑞𝑞 = 𝑘𝑘𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ∗ Δ𝑐𝑐 Overall Overall transfer coef. k Heat transfer k mass transfer 1 1 𝑠𝑠 1 1 1 𝐾𝐾 ∗ = + + = + 𝑘𝑘 𝛼𝛼1 𝜆𝜆 𝛼𝛼2 𝑘𝑘 𝛽𝛽𝐼𝐼 𝛽𝛽𝐼𝐼𝐼𝐼 K* = Gleichgewichtskonstante Analogue quantity Heat transport Mass transport Free flow 𝑔𝑔 ∗ 𝜅𝜅 ∗ Δ𝑇𝑇 ∗ 𝑙𝑙3 𝑔𝑔 ∗ Δ𝜌𝜌 ∗ 𝑙𝑙 3 𝐺𝐺𝑟𝑟𝑊𝑊𝑊 = 𝐺𝐺𝑟𝑟𝑆𝑆𝑆𝑆𝑆 = 𝜌𝜌 ∗ 𝜈𝜈 2 𝜈𝜈 2 κ = therm. expansion coefficient 𝑤𝑤 ∗ 𝑙𝑙 𝑤𝑤 ∗ 𝑙𝑙 Forced flow Re = Re = 𝜈𝜈 𝜈𝜈 𝜈𝜈 𝜈𝜈 Material props. Pr = Pr′ = 𝑆𝑆𝑆𝑆 = 𝐷𝐷 𝑎𝑎 (Prandtl/Schmidt) Lewis-Number 𝑆𝑆𝑆𝑆 Coupled heat and mass transport 𝐿𝐿𝐿𝐿 = Pr 𝑎𝑎 𝐷𝐷 Analogue quantity Heat transport Mass transport Nusselt-Number Sherwood Number Definition 𝛼𝛼 ⋅ 𝑙𝑙 𝛽𝛽 ⋅ 𝑙𝑙 𝑁𝑁𝑁𝑁 = 𝑆𝑆𝑆 = 𝜆𝜆 𝐷𝐷 Free flow 𝑁𝑁𝑁𝑁 = 𝑓𝑓(𝐺𝐺𝑟𝑟𝑊𝑊𝑊 , Pr) 𝑆𝑆𝑆 = 𝑓𝑓(𝐺𝐺𝑟𝑟𝑆𝑆𝑆𝑆𝑆 , 𝑆𝑆𝑆𝑆) Forced flow 𝑁𝑁𝑁𝑁 = 𝑓𝑓(Re, Pr) 𝑆𝑆𝑆 = 𝑓𝑓(Re, 𝑆𝑆𝑆𝑆) 𝑁𝑁𝑁𝑁 = 𝑎𝑎1 ∗ Re𝑎𝑎2 ∗ Pr 𝑎𝑎3 𝑆𝑆𝑆 = 𝑎𝑎1 ∗ Re𝑎𝑎2 ∗ 𝑆𝑆𝑐𝑐 𝑎𝑎3 a1, a2, a3 = Anpassungsparameter aus Experimenten ermittelt Lewis number: Division of NUSSELT-number by SHERWOOD-number: 𝑎𝑎3 𝑎𝑎2 𝑎𝑎3 𝑎𝑎𝑎 𝜈𝜈 𝑎𝑎3 𝑁𝑁𝑁𝑁 𝑎𝑎1 ∗ Re ∗ Pr Pr 𝑎𝑎 𝐷𝐷 = = = 𝜈𝜈 = 𝑆𝑆𝑆 𝑎𝑎1 ∗ Re𝑎𝑎2 ∗ 𝑆𝑆𝑐𝑐 𝑎𝑎3 𝑆𝑆𝑆𝑆 𝑎𝑎 𝐷𝐷 𝑆𝑆𝑆𝑆 𝑎𝑎 𝛼𝛼 𝑙𝑙 𝛽𝛽 𝑙𝑙 with: 𝐿𝐿𝐿𝐿 = = und 𝑁𝑁𝑁𝑁 = und 𝑆𝑆𝑆 = Pr 𝐷𝐷 𝜆𝜆 𝐷𝐷 𝑎𝑎3 𝜆𝜆 𝑁𝑁𝑁𝑁 𝛼𝛼 𝑙𝑙 𝐷𝐷 𝐷𝐷 1 = = = 𝑎𝑎 mit 𝑎𝑎 = 𝑏𝑏𝑏𝑏𝑏𝑏. 𝜆𝜆 = 𝑎𝑎 ⋅ 𝑐𝑐𝑝𝑝 ⋅ 𝜌𝜌 𝑆𝑆𝑆 𝜆𝜆 𝛽𝛽 ∗ 𝑙𝑙 𝑎𝑎 𝐿𝐿𝑒𝑒 3 𝑐𝑐𝑝𝑝 ⋅ 𝜌𝜌 𝑎𝑎3 −1 𝛼𝛼 𝐷𝐷 1−𝑎𝑎3 Gas (Water in air) Water Water in oil = 𝑐𝑐𝑝𝑝 𝜌𝜌 = 𝑐𝑐𝑝𝑝 ∗ 𝜌𝜌 ∗ 𝐿𝐿𝐿𝐿 𝛽𝛽 𝑎𝑎 a 1,94E-05 1,32E-07 9,98E-08 m2/s Lambda 2,50E-02 5,50E-01 1,50E-01 W/(m K) rho 1,29E+00 1,00E+03 9,00E+02 kg/m3 Approximatively: cp 1,00E+03 4,18E+03 1,67E+03 J/kg/K D 2,30E-05 5,00E-10 1,00E-11 m2/s 𝛼𝛼 𝑎𝑎 Le 8,43E-01 2,63E+02 9,98E+03 - ≈ 𝑐𝑐𝑝𝑝 ⋅ 𝜌𝜌 ≈ 𝑐𝑐𝑝𝑝 ⋅ 𝜌𝜌 𝛽𝛽 𝐷𝐷 Le^0,5 9,18E-01 1,62E+01 9,99E+01 - Le^0,33 9,58E-01 6,29E+00 2,09E+01 - Main diffusion conditions 1. Equimolar Diffusion: Exchange of similiar amount (volume) of substances in inverse directions of two components 2. Single sided diffusion Einseitige Diffusion One sinded exchange of one component through a semi-permeable wall 3. Non equimolar diffusion Exchange of any non equal amount (volume) of substances in inverse directions of two components 4. Multicomponent diffusion – more then two components One sided/ Non equimolar diffusion: „Stefan – Strom!“ = > Equimolar at ideal gases means equichor (same volume) = > Single side, non equi-molar => compensating volume flow Enhancing / decreasing? classical examples : Drying and BOUDOUARD-reaction(CO2+C = 2CO) Analogy Heat and Masstransport Exponent n=a3: acc. to Mersmann (1980) Flow regime Purely laminar Turbulent with laminar boundary layer Start up process in Frictionless flow Turbulent flow Friction less flow (Re -> ∞)  Stefan Strom: Importance p p − pv,e 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡 f = ln pv, S − pv,e p − pv,S pv(T) l varied, ptotal Parameter 5 4,5 4 3,5 3 Stefan factor 2,5 500 mbar 1 bar 2 50 mbar 1,5 1 0,5 0 0 200 400 600 800 1000 1200 Vapour pressure  𝑛𝑛−1 𝛼𝛼 𝐷𝐷 𝑝𝑝𝑖𝑖𝑖𝑖 1−𝑛𝑛 𝑝𝑝𝑖𝑖𝑖𝑖 = 𝑐𝑐𝑝𝑝 ∗ 𝜌𝜌 ∗ 1− = 𝑐𝑐𝑝𝑝 ∗ 𝜌𝜌 ∗ 𝐿𝐿𝐿𝐿 1− 𝛽𝛽 𝑎𝑎 𝑝𝑝 𝑝𝑝 𝑝𝑝𝑖𝑖𝑖𝑖 − 𝑝𝑝𝑖𝑖𝑖 𝑝𝑝𝑖𝑖𝑖𝑖 + 𝑝𝑝𝑖𝑖𝑖 Exponent n=a3: acc. to Mersmann 𝑝𝑝 − 𝑝𝑝𝑖𝑖𝑖 ≈ 𝑝𝑝 − 2 =: 𝑝𝑝 − 𝑝𝑝𝑖𝑖𝑖𝑖 ln 𝑝𝑝 − 𝑝𝑝𝑖𝑖𝑖𝑖 Flow regime Purely laminar Turbulent with laminar boundary layer Start up process in Frictionless flow Turbulent flow Friction less flow (Re -> ∞) Summary Analogy between Heat and Mass Transfer  Heat Transfer Mass transfer Heat conduction 𝑝𝑝𝑖𝑖 ℜ 𝑞𝑞̇ = −𝜆𝜆 ⋅ 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝜐𝜐 𝑚𝑚̇ = −𝐷𝐷 ⋅ 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 ; 𝑅𝑅 = Diffusion 𝑅𝑅𝑅𝑅 𝑀𝑀 1 dim. 𝑑𝑑𝑑𝑑 𝐷𝐷 𝑑𝑑𝑝𝑝𝑖𝑖 𝑞𝑞̇ = −𝜆𝜆 ⋅ 𝑚𝑚̇ = − 𝑑𝑑𝑑𝑑 𝑅𝑅𝑅𝑅 𝑑𝑑𝑑𝑑 Transfer 𝛽𝛽 𝑞𝑞̇ = 𝛼𝛼 ⋅ 𝜐𝜐𝑓𝑓 − 𝜐𝜐0 𝑚𝑚̇ = ⋅ 𝑝𝑝𝑖𝑖0 − 𝑝𝑝𝑖𝑖 = 𝛽𝛽 ⋅ 𝑐𝑐0𝑖𝑖 − 𝑐𝑐𝑖𝑖 𝑅𝑅𝑅𝑅 Balance Δ𝑝𝑝𝑖𝑖 𝑄𝑄̇ = 𝛼𝛼𝛼𝛼 Δ𝜐𝜐 𝑚𝑚 𝑀𝑀̇ = 𝛽𝛽𝛽𝛽 = 𝛽𝛽𝛽𝛽 Δ𝑐𝑐𝑖𝑖 m 𝑅𝑅𝑅𝑅 m Log. average Δ𝜐𝜐 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 − Δ𝜐𝜐 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 Δ𝑐𝑐𝑖𝑖 − Δ𝑐𝑐𝑖𝑖 Difference ℎ𝑢𝑢𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Δ𝑐𝑐𝑖𝑖 Δ𝜐𝜐 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 ln ℎ𝑢𝑢𝑢𝑢𝑢𝑢 ln Δ𝑐𝑐𝑖𝑖 Δ𝜐𝜐 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆 Heat conduction; 𝑞𝑞̇ Aera specific heat flux, 𝜐𝜐 Temperature, 𝑚𝑚̇ spec. Mass flux, 𝐷𝐷 Diffusion coefficient (T ≈ const. ) 0 Surface, f Fluid, A Boundary area , i Component i;𝑝𝑝𝑖𝑖0 equilibrium partial vapour pressure pi Partial vapour pressure, c Concentration  Convection Heat Transfer Mass transfer Reynolds number 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢𝑢𝑢 Re = = Re = = 𝜈𝜈 𝜂𝜂 𝜈𝜈 𝜂𝜂 Dimensionless 𝛼𝛼𝛼𝛼 𝛽𝛽𝛽𝛽 Transfer coeff. 𝑁𝑁𝑁𝑁 = ; Nusselt 𝑆𝑆𝑆 = ; Sherwood 𝜆𝜆 𝐷𝐷 Flux ratios 𝜈𝜈 𝜂𝜂𝜂𝜂𝜂𝜂 𝜂𝜂𝜂𝜂 𝜈𝜈 Pr = = = ; Prandtl 𝑆𝑆𝑆𝑆 = ; 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 Momentum / heat 𝑎𝑎 𝜌𝜌𝜌𝜌 𝜆𝜆 𝐷𝐷 /Diffusion Ratio Sc/ Pr 𝑆𝑆𝑆𝑆 𝐿𝐿𝐿𝐿 = 𝑎𝑎 Pr Lewis 𝐷𝐷 Nusselt/ Sherwood 𝑁𝑁𝑁𝑁 = 𝑘𝑘 Re𝑏𝑏 Pr 𝑐𝑐 ; 𝑆𝑆𝑆 = 𝑘𝑘 Re𝑏𝑏 𝑆𝑆 𝑐𝑐 𝑐𝑐 ; Function k, b, c Constants k, b, c Constants