MECH-414 Momentum & Energy Transport ANALYSIS OF FLOW IN PIPES AND OVER SURFACES 1. Introduction The Navier-Stokes equations govern the motion of a Newtonian fluid; see page 214, Fluid Mechanics Text by Fox, McDonald and Pritchard, 6th Edition. When nondi
The Thermal Boundary Layer The thermal boundary layer is the thin region of wall-adjacent viscous flow in which temperature gradients exist. Consider Ts = Constant. Then, the thickness of the thermal boundarylayer is the normal distance from the surface t
1
Free Convection (or, Natural Convection) No externally forced flow Buoyancy (density differences) cause flow to occur < 0 , i.e., T . T
Need temperature gradient, in a gravity field.
T1
T1
flow
T2 T 2 > T1
Sufficiently large to overcome retarding influ
Non-dimensionalization of Momentum Equation
u u 2u u +v = g (T T ) + 2 x y y
Introduce non-dimensional quantities with superscript *:
u = u * U o ; x = x * L ; y = y* L T = T + T* (Ts T )
Note that all reference quantities are always dimensional.
LHS:
v*
Boundary-Layer Thickness,
This is a very thin layer in the immediate neighborhood of the body where the fluid viscosity exerts an essential influence, in the sense that the shearing stresses are important here.
= y |u = o.99U e
Displacement Thickness, *
Notes on Boundary Layers, Integral Approach, etc.
o Flat Plate self-similar governing differential equation:
2f + ff = 0, where f ( x ) ~ ( x, y ) , = ( x, y ) .
o Pipe flow - fully developed governing differential equation:
u = constant, u =
u* r* , r= u
Film Condensation on a vertical wall
g Ts T T
Ts T Tsat T
Ts < Tsat < T
T
Ts < T
l
u
Without Phase Change With Phase Change
u,v,T ul ,vl ,Tl
Governing Differential Equations and Boundary Conditions Momentum:
Continuity:
u + v u = g + 2 u g) x y ( y2 u + v
Chapter 10 Summary of Concepts
1. Pool Boiling pg. 596 The liquid is quiescent, and its motion near the surface is due to free convection and due to mixing induced by bubble growth and detachment. Forced Convection Boiling pg. 596
Fluid motion is induced
February 1, 2006 What is the Hydrodynamic Boundary Layer? The hydrodynamic Boundary Layer on a flat plate is a thin region adjacent to the surface (boundary) within which viscous effects are confined. How is the thickness of the Hydrodynamic Boundary Laye
Self-Similarity
A flow is said to be self similar if the prevailing velocity profiles at various streamwise stations x can be made congruent, through the use of appropriately defined independent and dependent variables. The transformation to these variabl
Boundary-Layer Theory
Simplified Navier-Stokes Equations for a very Thin Layer of Flow Adjoining a Solid Boundary
Laminar Boundary Layer Over Flat Plate
2-D Incompressible Flow, x-y coordinates, Navier-Stokes Equations:
u u x + v u y = p x + (u xx + u yy
Summary of Thermal Considerations for Pipe Flows q s = constant h fd hD Nu D @ k h fd = constant Nu D = constant
Ts = constant h fd = constant Nu D = constant
T . x
dependence on r, x independent of r
depends on r
d Tm = dx
dT = independent of r dx
T Ts
SOLUTION FOR THE BOUNDARY LAYER ON A FLAT PLATE
Consider the following scenario. 1. A steady potential flow has constant velocity U in the x direction. 1. An infinitely thin flat plate is placed into this flow so that the plate is parallel to the potentia
Introduction to Convection: Introduction Flow and Thermal Considerations
Chapter Six and Appendix E Sections 6.1 to 6.9 and E.1 to E.3
Boundary Layer Features Bounda
Boundary Layers: Physical Features
Velocity Boundary Layer A consequence of viscous effe
Introduction to Convection: Introduction Flow and Thermal Considerations
Chapter Six and Appendix E Sections 6.1 to 6.9 and E.1 to E.3
Boundary Layer Features Boundary
Boundary Layers: Physical Features
Velocity Boundary Layer A consequence of viscous ef
1
Pool Boiling: Boiling in an otherwise Stationary Liquid
liquid: boiling
Ts > Tsat
q = h (Ts Tsat ) s
(10.3)
= h ( Te )
excess temperature
Boiling Curve
Nucleate Boiling
q = C ( Te ) a [n ] b s
(10.4)
= C( Te ) 1.2 ( Te ) 5 or 6 1/ 3
q ( Te ) 3 s
See E