Boundary Layer Similarity

Boundary Layer Similarity

• B. L. Equations:

≈0

≈0

For low-speed, forced convection flows :

RHS = diffusion terms

LHS = convection terms

The last three equations are very similar

B.L. Similarity Parameters: Re, Pr, Sc

L - is some characteristic length

U_∞ - is the free stream velocity

T_∞ - is the free stream temperature

T_S - is the surface temperature

C_A,S is the concentration of species A at the surface

C_A,∞ - is the concentration of species A free stream

Using the above dimensionless independent variables, the B.L. equations become:

≈0

Reynolds #:

Prandtl #: Pr≈1 for gases

Pr>>1 for oils

Pr<<1 for liquid metane

Schmidth #:

Lewis #:

Other dimensionless groups:

Biot #:

Nusselt #:

Stanton #:

Sherwood #:

Mass Transfer

Stanton #:

Functional Form of the solutions for the B.L. Equations

Eq. 2 :

≈ on the geometry of the surface

universally applicable to different fluids and over a wide range of values for U_∞ and L

Eq. 3 :

Eq. 4:

The similarity parameters reduce the independent variables and make the solutions universally applicable

Boundary Layer Analogies

• provide relations between C_f, Nu and Sh

The Heat and Mass Transfer Analogy

Eq. (3) and (4), as well as the associated boundary conditions are analogous → T^* and C^*_A must be of the same functional form

└→ heat and mass transfer relations for a particular geometry are interchangeable

Hence, h can be obtained from knowledge of h_m and vice-versa.

The same relation can be applied to and , and it may be used in turbulent, as well as laminar flow.

Example

T_∞=20⁰C

V=100 m/s T_S=80⁰C

P=1 at

L=1m

• The Reynolds Analog

If and Sc=Pr => Eqs. (2)-(4) are of precisely the same form

u^*,T^* and C_A^* are equivalent as solutions.

Reynolds analogy

It has been shown that, with certain corrections, the analogy may be applied over a wide range of Pr and Sc.

modified Reynolds or Chilton-Colburn analogies

!!! also valid for averages:

End Note:

• T^* and C_A^* - fields may be obtained analytically only for a few simple cases: h→h_m
• The more practical approach frequently involves calculating h and h_m from empirical equations; the empirical equations are obtained by correlating measured convection and mass transfer results in terms of appropriate dimensionless groups

PROBLEM 1.

KNOWN: Air flow conditions and drag force associated with a heater of prescribed surface temperature and area,

FIND: Required heater power.

SCHEMATIC:

ASSUMPTIONS: (1) Steady-state conditions, (2) Reynolds analogy is applicable, (3) Bottom surface is adiabatic.

PROPERTIES: Air (T_f=350K, 1 at): ρ=0.995 kg/m³, c_p=1009 J/Kg K, Pr=0.7

ANALYSIS: The average shear stress and friction coefficient are

From the Reynolds analogy, modified:

Solving for and substituting numerical values, find

=0.995 kg/m³(15 m/s) 1009 J/kg K (8.93x10^-3/2)(0.7)^-2/3 ;

=85 W/m²K

Hence, the heat rate is :

q=A(T_S-T )=85W/m²K(0.25m²)(140-15)^oC;

q=2.66kW

COMMENTS: Due to bottom heat losses, which have been assumed negligible, the actual power requirement would exceed 2.66kW.