Convection is the mechanism of heat transfer through that occurs between a solid surface & the surrounding fluid in the presence of bulk fluid motion.
Convection is classified as Natural (or free) and Forced convection depending on how the fluid motion is initiated.
The rate of convection heat transfer is expressed by Newton’s law of cooling and it is given as Forced Convection
Where h is the heat transfer coefficient and A is the Area of the contact.
The convective heat transfer coefficient h strongly depends on the fluid properties and roughness of the solid surface, and the type of the fluid flow (laminar or turbulent).
h = f(ρ, V, D, μ, Cp, K)
BOUNDARY LAYER
HYDRODYNAMIC BOUNDARY LAYER
Hydrodynamic Boundary Layer is defined as a thin region formed on the plate inside which velocity gradients are seen in a normal direction to the plate.
(at any x measured from leading edge)
If Rex < 5 × 105, then flow is LAMINAR
If Re= > 6.5 to 7 × 105, Then flow is TURBULENT
CASE 1: Flow over flat plates
Hydrodynamic boundary layer
δ → Thickness of Hydrodynamic Boundary Layer at any distance x measured from leading edge [i.e., x = 0] of the plate.
δ = f(x)
THERMAL BOUNDARY LAYER
Similar to velocity boundary layer, a thermal boundary layer develops when a fluid at specific temperature flows over a surface which is at different temperature.
T = f (x,y)
δt = Thickness of thermal Boundary Layer
δt = f(x)
The Thermal boundary layer thickness is defined as the distance measured form the solid boundary in y direction at which
Thermal boundary layer (T∞ > Tw)
ENERGY BALANCE FOR THERMAL BOUNDARY LAYER
Energy balance for thermal boundary layer in a control volume
T = f (x, y)
Assuming steady state conditions for the control volume,
dq = Heat conducted through the stagnant fluid layer at y = 0 = Heat convected between the plate & free stream fluid
Where, hx = Local convective heat transfer coefficient at that x,
variation of hx with x
NUSSELT NUMBER (Nu)
PRANDTL NUMBER (Pr)
peclet Number-
it is the product of Reynold number and prandtl number.
Peclet number = Reynold number × Prandtl number
Stanton Number -
it is the ratio of nusselt number to the peclet number.
Relation between thermal boundary layer and hydrodynamic boundary layer
Reynold colburn analogy
Flow Over Flat Plate
The friction and heat transfer coefficient for a flat plate can be determined by solving the conservation of mass, momentum, and energy equations (either approximately or numerically).
Nusselt number can be expressed as:
where C, m, and n are constants and L is the length of the flat plate.
The properties of the fluid are usually evaluated at the film temperature defined as:
Laminar Flow
Local Nusselt number at the location x for laminar flow over a flat plate are
Average Nusselt number for entire length of the plate
The local friction coefficient at the location x for laminar flow over a flat plate are
where x is the distance from the leading edge of the plate and
Average local friction coefficient for entire length of the plate
Critical Reynolds number for flow over flat plate is 5×105
Turbulent Flow
Local Nusselt number at location x for turbulent flow over a flat isothermal plate are:
Average Nusselt number
Local friction coefficient
Average friction coefficient
Pipe Flow
LAMINAR FLOW
The bulk mean temperature (Tm) at a given cross-section of the pipe of flowing fluid is defined as the constant temperature which takes into account the variation of temperature of fluid layers with respect to radius at that cross-section of the pipe and hence indicates the total thermal energy or enthalpy carried by the fluid through that cross-section.
Bulk Mean Temperature
CASE 1: CONSTANT HEAT FLUX
When the fluid flowing in the pipe is subjected to constant heat flux
Constant Heat flux
No dependence on Re and Pr.
CASE 2: CONSTANT SURFACE TEMPERATURE
When the fluid flowing in the pipe is in contact with the surface having constant temperature.
Constant Surface Temperature
TURBULENT FLOW
n = 0.3 for cooling
n = 0.4 for heating
FREE CONVECTION OR NATURAL CONVECTION
No velocity evident but the flow occurs due to buoyancy forces arising out of density changes of fluid.
Free convection over different geometries
In any free convection heat transfer,
h = f (g, β, ΔT, L, μ, ρ, Cp, k)
μ, ρ, Cp, k = Thermophysical Properties of fluid
β = Isobaric volume expansion coefficient of fluid [per kelvin]
Characteristic dimension for various dimensionless numbers:
For vertical plates & vertical cylinders.
For horizontal plate for horizontal cylinder
Characteristic dimension for various geometries
GRASHOF NUMBER(Gr)
where
g = gravitational acceleration, m/s2
β = coefficient of volume expansion, 1/K
δ = characteristic length of the geometry, m
ν = kinematics viscosity of the fluid, m2/s
Grashof Number(Gr) replaces Reynolds Number (Re) in free convection heat transfer.
Therefore, in any free convection heat transfer,
Heat transfer coefficient
Nu = f (Gr Pr)
product of Gr & Pr is called Rayleigh Number (Ra)
Nu = C (Gr×Pr)m
C & m are constants which vary from case to case.
m = 1/4 for Laminar Flow
m = 1/3 for Turbulent Flow
The flow during free convection heat transfer is decided as Laminar or Turbulent based on the value of (Gr Pr) product.
If Gr Pr < 109 Then, flow is Laminar
If Gr Pr > 109 Then, flow is Turbulent
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