Seth Jai Parkash Mukund Lal Institute of Engineering & Technology, Radaur.
Heat ,mass and momentum transfer
Unit process can be classified into three fundamental transfer processes.
General molecular transport equation:
All three of the molecular transport process of momentum, mass and heat are characterized by the same general type of equation.
Molecular diffusion equation for momentum :
ح =momentum transferd/s.m 2
μ / ρ is kinematic viscosity in m 2 /s
z is the distance in m
v x ρ is momentum/m 3
where the momentum has the units of kg. m /s.
Molecular diffusion for heat conduction for constant c p and ρ
q/A is heat flux in W/m 2
α is thermal diffusivity in m 2 /s
ρ c p T is J/m 3
Molecular diffusion equation for mass transfer
j AZ is molar flux of component A due to molecular diffusion in
kg mol A/ s.m 2
D AB is the molecular diffusivity of molecules A in B in m 2 /s
C A is the concentration of A in kg mol/m 3
z is the distance o f diffusion in meters
Analogies All three molecular transport equations are identical. There is mathematical analogy between these equation but the actual physical mechanism occurring is totally different .E.g. In the mass transport two components are being transported by relative motion .In heat transfer, molecules are relatively stationary and transport is taken by electrons. Transport of momentum is occurred by several types of mechanism.
All the fluxes are on the left hand side of
three equations. All fluxes have the same
units i.e. quantity transferred /per unit time
per unit area .
The transport properties μ / ρ , α and D AB have the units of m 2 /s.All concentration are represented as momentum/m 3 , j/m 3 , kgmol/m 3 .
Since basic mechanism of heat, mass and momentum transport is essentially same, it is some times possible to directly relate heat transfer coefficients, mass transfer coefficients and friction factors by means of analogies.
Analogy involving momentum transfer is only valid if there is no form drag, hence these are limited to flow over flat plates and inside conduits .
Turbulent diffusion equation for momentum, heat and mass transfer
The most simple and crude analogy in turbulent diffusion equations is that turbulent eddies transport or eddies diffusivities are same for all modes of transport.
Momentum eddy diffusivity Thermal eddy diffusivity Mass eddy diffusivity
Another analogy probably the oldest is the
“ Reynolds Analogy”. This relates fanning friction factor for fluid flow to heat transfer.
Fanning friction factor can be defined as shear stress at the surface divided by the product of density times velocity head.
Fanning friction factor
For momentum transfer
For fluid flow in a pipe, heat transfer equation from fluid to wall can be written as
After dividing the momentum transfer equation with heat transfer equation and by assuming thermal diffusivity and momentum diffusivity negligible and equal eddy diffusivities we get a equation
Analogies Integrating between the T=Ti and v=0 to some point where T is the same as the bulk T and assume that velocity at this point is same as average velocity.
Analogies This become Reynolds Analogy. It postulates direct interaction between the turbulent core of the flow and the walls.
Right hand side term in the Reynolds analogy is the Stanton number. Stanton number is a dimensionless group made up of other more familiar groups. Reynolds analogy gives reasonable values for gases where Prandtl number is roughly one.
Nusslet number: It establishes the relation between convective film coefficient h, thermal conductivity of fluid k and length parameter d of a physical system.
It is also interpreted as the ratio of temperature gradient to an overall reference temperature gradient.
Reynold number: It is ratio of inertial force to viscous forces in fluid motion. At low Reynold numbers viscous effects dominate and fluid motion is laminar. At high Reynold numbers inertial effects leads to turbulent flow.
N Re < 2100
N Re > 4000
Laminar flow Turbulent flow
Prandtl number: It is ratio of kinematic viscosity to thermal diffusivity of a fluid. It indicates the relative ability of the fluid to diffuse momentum and internal energy by molecular mechanism. Prandtl number is connecting link between velocity field field and the temperature field and its value strongly influences relative growth of velocity and thermal boundary layers.
Schmidt number: It is the ratio of shear component of diffusivity μ / ρ to the diffusivity of mass transfer. Values of Schmidt number for gases range from about 0.5 to 2. For liquids Schmidt numbers range from about 100 to over 10000 for viscous liquids.
Reynolds analogy breakdown if viscous sub layer become important, because eddy diffusivities diminish to zero and molecular diffusivities becomes important.
Prandtl modified the Reynolds analogy by writing regular diffusion equations for viscous sub layer and Reynolds analogy equation for turbulent core region.
Viscous sub layer is the region where the velocity is proportional to distance from the wall. The second region called buffer layer is the region of transition between the viscous sub layer with practically no eddy activity. Third region called turbulent core has violent eddy activities
Boundary layer in pipe Viscous sub layer Buffer zone Turbulent core
Viscous sub layer Buffer layer Turbulent core y + V + Dimensionless velocity ratio Dimensionless number
If the laminar sub layer is included in Reynolds analogy, the Prandtl analogy applies. Prandtl analogy included the ratio of mean velocities in laminar sub layer and core as well as the Prandtl number for heat transfer. If the Prandtl number is reduces to one it becomes Reynolds analogy.
Von korman modified the Prandtl analogy by considering buffer layer in addition to viscous sub layer and turbulent core.
Chilton and Colburn J-factor analogy is most successful and widely used analogy. This analogy is based on correlations and data rather than assumptions about transport mechanism.
Chilton Colburn j-factor analogy is simple analogy. This analogy is valid for turbulent flow in conduits with N Re > 10000, 0.7< N Pr > 160