Sample Problem Associated to Transport Phenomena
Sample Problem Associated to Transport Phenomena At the fluid dynamic laboratory, a group of engineers are doing a test. Air is at 300°K, 1 atmospheric pressure flows at 200 Km/h over a flat surface 1 m long. The engineers consider the concentration of vapor in the air is negligible. The diffusion of water vapor into air is 0.5 x 10β4 m²/s and the Colburn J factor for mass transfer coefficient is given by JM = 0.03 x π π β0.2 . Calculate the mass transfer coefficient (βπ) of water vapor from the flat surface into the air. Properties of air at 300°K and 1 atmosphere are: π = 1.177 Kg/m³ Cp = 1005 J/Kg°K π = 1.84 π₯ 10β5 Kg/m.s Pr (Prandtl) = 0.708 0.173 m/s b) 0.841 m/s c) 0.542 m/s d) 1.533 m/s Solution To solve this problem , the first step is to identify the System and Process. Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 1 Identifying our System Air (at 300°K, 1 atmospheric pressure) flows at 200 Km/h over a flat surface (which contain Water Vapor) 1 m long Our system contains two components (Air and Water Vapor) whose concentration vary from point to point. Therefore, there is a natural tendency for mass to be transferred, minimizing the concentration differences within the system. The transport of one constituent from a region of higher concentration to that of lower concentration is called mass transfer In this case Water Vapor is diffusing into Air Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 2 IMPORTANT CONCEPT The Mass transfer in our system is due to convection and involves transfer of mass between a moving fluid (Air) and a flat surface (which contain Water Vapor) Similar to Newtonβs law for convective heat transfer, the convective mass transfer equation can be written as: Our objective (Unknown) π = hm x A x βπ where hm is the convective mass transfer coefficient and Ξc is the difference between the boundary surface concentration and the average concentration of fluid stream Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 3 Similar to convective heat transfer, convective mass transfer coefficient depends on the type of flow, i.e., laminar or turbulent and forced or free. In general the mass transfer coefficient is a function of the system geometry, fluid and flow properties and the concentration difference. In convective mass transfer the non-dimensional numbers corresponding to Prandtl and Nusselt numbers of convective heat transfer are called as Schmidt and Sherwood numbers. These are defined as: ππ π π³ Sherwood Number (Sh) = π«π Long or diameter (In our case is the longitude of flat surface) Diffusion coefficient(Diffusivity) Schmidt Number (Sc) = π π«π kinematic viscosity Diffusion coefficient(Diffusivity) Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 4 Very important concept The first recognition of the analogous behaviour of mass, heat and momentum transfer was reported by Osborne Reynolds in 1874. Reynolds postulated that the mechanisms for transfer of momentum, energy and mass are identical. Accordingly: βπ= βπ πβ β = πβΞ½ββπΆπ = π π Convective Mass Transfer Coefficient β= Heat Transfer Coefficient π= Moody friction factor Vβ = Velocity of free stream The Reynolds analogy is interesting because it suggests a very simple relation between different transport phenomena. This relation is found to be accurate when Prandtl and Schmidt numbers are equal This is applicable for mass transfer by means of turbulent eddies in gases. to one (1) . In this situation, we can estimate mass transfer coefficients from heat transfer coefficients or from friction factors. Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 In our problem Prandtl (Pr) = 0.708 5 we cannot apply Reynolds Analogy Therefore Chilton β Colburn Analogy Chilton and Colburn, using experimental data, sought modifications to the Reynoldβs analogy that would not have the restrictions that Prandtl and Schmidt numbers must be equal to one. They defined for the j factor for mass transfer as JM = πβ β ππ 2/3 = π (Fanning Friction Factor) 2 JM= Chilton-Colburn factor for mass transfer In accordance with our case The analogy is valid between momentum and mass transfer for a flat plate: This analogy is valid for gases and liquids within the range of 0.6 < Sc < 2500 and 0.6 <Pr < 100 Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 6 The Chilton-Colburn analogy has been observed to hold quite well in laminar or Turbulent over flow over plane surfaces But this is not always the case for internal flow and flow over Irregular geometries, and in such cases specific relationships developed should be used π= 4 x π(Fanning Friction Factor) Therefore JM = πβ β π = π(Fanning Friction Factor) 4 ππ 2/3 π = 8 Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 7 For Flat Plate JM = 0.664 π π β1/2 For Laminar flow JM = 0.037 π π β0.2 For Turbulent flow Now we need to verify if our flow is turbulent or laminar In order to know what JM factor formula we will use Re = πβπβπΏ π = 1.1744 β 55.5 β 1 = 3,539398 .79 β5 1.8463 β 10 (Turbulent) Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 8 We use the JM according with our data JM = 0.03 π π β0.2 = πβ β ππ 2/3 0.00147 (approx.) ππ ππ π π³ π«π Therefore = ππ π«π π³ π = π«π = π = πβπ·π 1.84β10β5 1.1774β0.5β10β4 = 0.3126 (approx.) = ππ = ππ β π ππ = π/π ππ π½ 1.47 β 10β3 β 55.5 = 0.175 m/s (approx.) = 2/3 0.3126 Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 9 Final Comments Mass transfer by convection involves the transport of material between a boundary surface (such as solid or liquid surface) and a moving fluid or between two relatively immiscible, moving fluids. There are two different cases of convective mass transfer: 1. Mass transfer takes place only in a single phase either to or from a phase boundary, as in sublimation of naphthalene (solid form) into the moving air. 2. Mass transfer takes place in the two contacting phases as in extraction and absorption. Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 10 Reynolds Analogy Reynolds Analogy is restricted to situations for which Pr = Sc = 1 When the molecular diffusivity of momentum, heat, and mass are equal to each other The velocity, temperature, and concentration boundary layers coincide In this case the normalized velocity, temperature and concentration profiles will coincide and the slope of these three curves at the surface will be identical Reynolds analogy enables us to determine the unrelated friction, heat transfer, and mass transfer coefficients when only one of them is known Laminar Transition Turbulent Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 11 Chilton β Colburn Analogy Valid when Pr β ππ β 1 When the friction or heat transfer coefficient is known, the mass transfer coefficient Can be determined directly from the Chilton-Colburn Analogy This analogy is valid for gases and liquids within the range of 0.6 < Sc < 2500 and 0.6 < Pr < 100. The power of the Chilton-Colburn j-factor analogy is represented various experimental values of JM or JH from a flat plate with flow parallel to the plate surface. The Chilton-Colburn analogy has been observed to hold for many different geometries. For example, flow over flat plates, flow in pipes, and flow around cylinders Carlos Batista M.Sc. P Eng. CCEA www.cbaengineer.com Phone: 403-245-8505 12
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