analogies among mass, heat, and momentum transfer
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Analogies among Mass, Heat, and Momentum Transfer
Analogies
Heat Mass (sometimes) Momentum
Analogies are useful tools
1. An aid to understand transfer phenomena2. A sound means to predict behavior of systems for
which limited quantitative data are available
Molecular Transport Equations
RECALL:driving force
rate of transport = resistance
( )xyx
d vdy
( c T)y pq d
A dy
* A
Ay AB
dcJ D
dy
MOMENTUM HEAT MASSNewton’s law Fourier’s law Fick’s law
Analogous quantities in transport phenomena
Reynolds Analogy
The general transport equation can be written in the form
𝜓=− (𝛿+𝐸 ) 𝑑 Γ𝑑𝑥
where ψ = flux of a property at any value of xδ = molecular diffusivityE = eddy diffusivityГ = volume concentration of transferent property
Turbulent diffusion equations
Transfer coefficient for momentum
Integrating the above equation and multiplying by A to get a rate equation,
where A = cross-sectional area perpendicular to flow = mean eddy diffusivity = ratio of the difference in concentration of transferent property between the wall and the mean value and the mean value of the fluid to the maximum difference between the wall and the center
D = diameter
In cylindrical geometry,
Transfer coefficient for momentum
(𝜓 𝐴 )=−4(𝛿+𝐸 )𝛾 𝐷
(Γ1−Γ ) 𝐴
The transfer coefficient is then defined as
𝜑=−4(𝛿+𝐸 )𝛾 𝐷
(𝜓 𝐴 )=−Γ 1− Γ
1𝜑 𝐴
Substituting and rearranging,
Transfer coefficient for momentum
(𝜓 𝐴 )=−4(𝛿+𝐸 )𝛾 𝐷
(Γ1−Γ ) 𝐴
The transfer coefficient is then defined as
𝜑=−4(𝛿+𝐸 )𝛾 𝐷
(𝜓 𝐴 )=−Γ 1− Γ
1𝜑 𝐴
Substituting and rearranging,
Transfer coefficient for momentum
(𝜓 𝐴 )=−Γ 1− Γ
1𝜑 𝐴
For momentum transfer,
Γ=𝜌𝑣𝜓=𝜏
𝜏=−𝜑 [𝜌𝑣1−𝜌 𝑣]
Transfer coefficient for momentum
𝜏=−𝜑 [𝜌𝑣1−𝜌 𝑣]
At the wall, v1 = 0 so that,
𝜏=𝜑𝜌𝑣
𝜑=𝜏𝜌𝑣
If we divide by ,
𝑓2
=𝜑𝑣
=𝜏𝜌𝑣2
The Reynolds analogy
For turbulent transport,For heat transfer,
For momentum transfer,
We assume that α and μ/ρ are negligible, and that
The Reynolds analogy
Dividing the momentum equation by the heat equation then gives
( 𝜏𝑞𝐴 )𝑐𝑝∫𝑇 𝑖
𝑇
𝑑𝑇=∫0
𝑣𝑎𝑣
𝑑𝑣
The Reynolds analogy
Substituting and
The Reynolds analogy
Stanton number
𝑁 𝑆𝑡=𝑁 𝑁𝑢
𝑁𝑅𝑒𝑁 𝑃𝑟
= f2
Dimensionless Groups
Dim. Group Ratio Equation
Prandtl, Pr molecular diffusivity of momentum / molecular diffusivity of heat
Schmidt, Sc momentum diffusivity/ mass diffusivity
Lewis, Le thermal diffusivity/ mass diffusivity
Stanton, St heat transferred/ thermal capacity
The Reynolds analogy
Experimental results show that the above equation1. Correlate data approximately for gases in turbulent flow2. DOES NOT correlate experimental data for liquids in turbulent flow3. DOES NOT correlate experimental data for any fluids in laminar flow
* 0.6 < NPr for gases < 2.5
It was concluded that the Reynolds analogy is valid ONLY at NPr = 1
f2=
hc p𝜌 𝑣
The Reynolds analogy
In a similar manner, we can relate mass transfer with momentum transfer
And the complete Reynolds analogy is
For turbulent transport
The Reynolds analogy
Experimental results show that the above equation1. Correlate data approximately for gases in turbulent flow2. DOES NOT correlate experimental data for liquids in turbulent flow3. DOES NOT correlate experimental data for any fluids in laminar flow
* NSc for gases ~ 1.0
It was concluded that the Reynolds analogy is valid ONLY at NSc = 1
f2=𝑘𝑐′
𝑣
The Reynolds analogy
CONCLUSIONS1. At NPr = NSc = 1, the mechanisms for mass, heat, and momentum are
identical2. For other fluids, transfer processes differ in some manner
functionally related to the Pr and Sc numbers.
The Reynolds analogy
Note that the Reynolds analogy assumes that 1. the turbulent diffusivities are equal and2. the molecular diffusivities are negligible.
1. For other fluids, where usually the case for liquids2. We CANNOT neglect molecular diffusivities in the boundary layer where diffusion, conduction, and viscosity are important
When are these assumptions not valid?
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