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Group Members2009-CH-2422009-CH-2272009-CH-2372009-CH-2032009-CH-229
Gulfam Raza Haidery2009-CH-242
Importance of dimensionless numbers in mass transfer
Contents:IntroductionPropertiesBuckingham π theoremDimensionless numbers in Mass TransferApplicationsPhysical significance
INTRODUCTION
In physics and mathematics, the dimension of a object is defined as the minimum number of coordinates needed to specify any point within it.
PROPERTIESQuantity with No Physical dimension but it can has units which are dimensionless.Quantities have same dimensions and their ratio makes them dimensionless.
Buckingham π theoremThe Buckingham π theorem of dimensional analysis is that the functional dependence between a certain number (say, n) of variables can be reduced by the number (say, k) of independent dimensions occurring in those variables to give a set of p = n − k independent, dimensionless quantities.
CONT…The power consumption of a stirrer is a function of the density the viscosity diameter speed So n = 5 variables are built up from k = 3 dimensions which are:Length: L (m)Time: T (s)Mass: M (kg). According to the π-theorem, the n = 5 variables can be reduced by
the k = 3 dimensions to form p = n − k = 5 − 3 = 2 independent dimensionless numbers which are, in case of the stirrer:
Reynolds number (a dimensionless number describing the fluid flow regime)
Power number (describing the stirrer and also involves the density of the fluid)
Basil Ahmad Raza2009-CH-227
About Mass Transfer
MT BIOT NUMBERRatio of Diffusive to Convective MT
Resistance
film mass transfer coefficientLC - characteristic length
DAB - mass diffusivity
Determines Uniformity of concentration in solids
MT FOURIER NUMBER Ratio of current time to time to reach steady
state Fo = Dt/L2
MT PRANDTL NUMBERRatio of momentum to specific diffusivity
Pr = ν/DDetermines the ratio of fluid/ Mass transfer
boundary layer thickness
ν : kinematic viscosity, ν = μ / ρ, (SI units : m2/s) α : thermal diffusivity, α = k / (ρcp), (SI units : m2/s) μ : dynamic viscosity, (SI units : Pa s = (N s)/m2) k: thermal conductivity, (SI units : W/(m K) ) cp : specific heat, (SI units : J/(kg K) ) ρ : density, (SI units : kg/m3 ).
K
C p
Pry diffusivit Thermal
y diffusivit Momentum
MUHAMMAD USMAN2009-CH-237
MT NUSSELT NUMBERRatio of length scale to diffusion boundary
layer thicknessNu = hdL/Dfluid
Used to calculate MT coefficient hd
MT GRASHOOF NUMBERRatio of natural convection buoyancy force to
viscous force Gr = g βCΔCL3/ν2
Controls the ratio of length scale to natural convection boundary layer
g = acceleration due to Earth's gravityβ = volumetric thermal expansion coefficient
(equal to approximately 1/T, for ideal fluids, where T is absolute temperature)
Ts = surface temperature
T∞ = bulk temperatureL = lengthD = diameterν = kinematic viscosity
LimitationsThe transition to turbulent flow occurs in the
range 108 < GrL < 109 for natural convection from vertical flat plates. At higher Grashof numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar.
M.M.KHAN2009-CH-203
MT PECLET NUMBERRatio of convective to diffusive mass transfer
in fluidsPe = Re.Pr
Used to determine plug flow/perfect mixing CSTR model validity
MT RALEIGH NUMBERRatio of natural convective to
diffusive mass transportRa = Gr.Pr
Determines the transition to turbulent
Importance of DNDimensionless numbers allow you to
compare two systems that are vastly different by combining the parameters of interest.
For example, the Reynolds number, Re = velocity * length / kinematic viscosity. If you are testing an airfoil with a particular Re, and you want to run a simulation on a scaled-down model (length is smaller), you could increase fluid velocity or lower kinematic viscosity (change fluids) or both to establish the same Re and ensure you are working under comparable circumstances.
FAISAL ALI 2009-CH-229
Archimedes Number (Ar)
This represents the ratio of buoyancy and inertial forces
Formula:
where:
• g = gravitational acceleration (9.81 m/s²),
• ρl = density of the fluid, kg / m3
• ρ = density of the body, kg / m3
• μ = dynamic viscosity, kg / sm
• L = characteristic length of body, m
Applicationo
Used when dealing with Gravitational settling of particle in fluidIn engineering applications the Péclet number is often very large.
In such situations, the dependency of the flow upon downstream locations is diminished, and variables in the flow tend to become 'one-way' properties. Thus, when modeling certain situations with high Péclet numbers, simpler computational models can be adopted.
A flow will often have different Péclet numbers for heat and mass. This can lead to the phenomenon of double diffusive convection.
In the context of particulate motion the Péclet numbers have also been called Brenner numbers, with symbol Br, in honor of Howard Brenner.
Physical significanceThis represents the ratio of buoyancy and
inertial forces, which stands in for the contribution of natural convection. When Ar >> 1, natural convection dominates and when Ar << 1, forced convection dominates.
S. no. Parameters Symbols (units) Primary dimensions
1 Significant length Lc (m) L
2 Fluid density ρ (kg/m3) ML-3
3 Fluid viscosity µ (kg/m.s) ML-1t-1
4 Temperature difference
ΔT (deg.C or K) T
5 Coefficient of volume expansion
β (1/K) T-1
6 Acceleration due to gravity
g (m/s2) Lt-2
7 Thermal conductivity of fluid
k (W/m.K) MLt-3T-1
8 Heat transfer coefficient
h (W/m2.K) Mt-3T-1
9 Specific heat of fluid Cp (J/kg.k) L2t-2T-1
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