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