Analysis of Convection Heat Transfer. P M V Subbarao Professor Mechanical Engineering Department IIT Delhi. Development of Design Rules …. Methods to evaluate convection heat transfer. Empirical (experimental) analysis
P M V Subbarao
Mechanical Engineering Department
Development of Design Rules …
Hot Wall & Cold Fluid
Cold Wall & Hot Fluid
Fluid particles in contact with the surface have zero velocity
u(y=0) = 0; no-slip boundary condition
Fluid particles in adjoining layers are retarded
δ(x): velocity boundary layer thickness
The local momentum flux (gain or loss) is defied by
Newton’s Law of Viscosity :
Momentum flux of far field stream:
The effect of solid boundary :
ratio of shear stress at wall/free stream Momentum flux
Fluid particles in contact with the surface attain thermal equilibrium
T(y=0) = Ts
Fluid particles transfer energy to adjoining layers
δT(x): thermal boundary layer thickness
Plate surface is warmer than the fluid (Ts> T∞)
Plate surface is cooler than the fluid (Ts< T∞)
At the surface, there is no fluid motion, heat transfer is only possible due to heat conduction. Thus, from the local heat flux:
This is the basic mechanism for heat transfer from solid to liquid or Vice versa.
The heat conducted into the fluid will further propagate into free stream fluid by convection alone.
Use of Newton’s Law of Cooling:
Temperature distribution in a boundary layer of a fluid depends on:
n Potential for diffusion of momentum change (Deficit or excess) created by a solid boundary.
a Potential for Diffusion of thermal changes created by a solid boundary.
Prandtl Number: The ratio of momentum diffusion to heat diffusion.
Other scales of reference:
Length of plate: L
Free stream velocity : u
Local Nusselt Number
First Law of Thermodynamics for A CV
Or Energy Equation for a CV
How to select A CV for External Flows ?
The rate of change of energy of a control mass should be equal to difference of work and heat transfers.
Energy equation per unit volume:
The net Rate of work done on the element is:
From Momentum equation: N S Equations
Substitute the work done by shear stress:
This is called the first law of thermodynamics for fluid motion.
Equation for Distribution of Temperature
Relative sizes of Momentum & Thermal Boundary Layers …
Liquid Metals: Pr <<< 1
Gases: Pr ~ 1.0
Water :2.0 < Pr < 7.0
Oils:Pr >> 1
Consider the flow over a parallel flat plate.
Assume two-dimensional, incompressible, steady flow with constant properties.
Neglect body forces and viscous dissipation.
The flow is nonreacting and there is no energy generation.
The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation:
Define characteristic parameters:
L : length
u∞: free stream velocity
T ∞: free stream temperature
x, y : positions (independent variables)
u, v : velocities (dependent variables)
T : temperature (dependent variable)
also, recall that momentum requires a pressure gradient for the movement of a fluid:
p : pressure (dependent variable)
Similarity parameters can be derived that relate one set of flow conditions to geometrically similar surfaces for a different set of flow conditions:
The velocity profile that satisfies above conditions:
Direction of similarity
One important consequence of this solution is that, for pr >0.6:
Local convection heat transfer coefficient:
For large Pr (oils):
For small Pr (liquid metals):
Pr > 1000
Pr < 0.1
Fluid viscosity less than thermal diffusivity
Fluid viscosity greater than thermal diffusivity
Has been developed by Churchill and Ozoe..
Local Nusselt number:
For a smooth flat plate: Rexc = 5 X 105
Valid over the ranges 10 < Rel < 107 and 0.6 < Pr < 1000
For triangular pitch:
Dsis the inner diameter of the shell.
Flow area associated with each tube bundle between baffles is:
where A s is the bundle cross flow area, Ds is the inner diameter of the shell, C is the clearance between adjacent tubes, and B is the baffle spacing.
Then the shell-side mass velocity is found with
Shell side Reynolds Number: