Heat Transfer

Differentiate between conduction, convection, and radiation

• Conduction: energy transfer via molecular interactions/electron transport without bulk motion: q = -kA dT/dx

  • ex) metal spoon getting hot in tea

• Convection: combined conduction + bulk fluid motion at a surface: q” = h(T_s - T_∞)

  • ex) wind cooling skin

• Radiation: electromagnetic emission due to temp → no medium required: q” = εσ(T_s^4 - T_sur^4)

  • feeling heat from sun

Define thermal conductivity and what does a high value imply?

• k [W/mK]: material property measuring ability to conduct heat

• High k = material transmits heat easily; small temp gradient needed for a given heat flux (metals > polymers/ceramics easily)

Difference between steady-state and transient heat transfer

• Steady-state: temperatures and heat rates do not change w/ time (∂T/∂t = 0).

  • ex) wall after long exposure to constant indoor/outdoor temps.

• Transient (unsteady): temps change with time (∂T/∂t ≠ 0)

  • ex) cooling of a hot object just placed in a cold room.

State Fourier’s Law and identify variables

q_x = -k A dT/dx: heat flows from hot to cold (negative sign)

• q_x: heat rate [W]

• A: area normal to x

• k: thermal conductivity

• dT/dx: temperature gradient

Define temperature gradient and relate to heat flow direction

• Temp gradient ∇T: spatial rate of change of temp

• Heat flux q'' = -k ∇T: direction of heat flow is opposite the gradient (from higher to lower temp)

Two flat metal plates are bolted together tightly, but microscopic air gaps exist at the interface. Which statement best describes the effect of these gaps?

They reduce the overall heat transfer rate by adding contact resistance

Microscopic gaps trap air (low k), introducing thermal contact resistance and reducing heat transfer

Water enters a circular tube whose walls are maintained at constant temperature at a specified flow rate and temperature. For fully developed turbulent flow, the Nusselt number can be determined from Nu = 0.023 Re0.8 Pr0.4. The correct temperature difference to use in the algebraic (integrated) form of Newton’s law of cooing (q = hA∆T) in this case is

The log mean temperature difference

For the same initial conditions, one can expect the laminar thermal and momentum boundary layers on a flat plate to have the same thickness when the Prandtl number of the flowing fluid is

Approximately one

Given similar initial conditions for laminar boundary layers forming during flow over a flat plate (one thermal, one velocity), the velocity boundary layer is likely to be much thicker than the thermal boundary layer at a given downstream location if the Prandtl number of the fluid is

large (»1)

What’s Nu?

a. hD/k (for flow over a cylinder of diameter, D)

b. A function of Re and Pr for a given geometry

c. a nondimensional form of the convection coefficient

Velocity (Momentum Boundary Layer

• a consequence of viscous effects associated w/ relative motion between a fluid & a surface

• a region of the flow characterized by shear stresses & velocity gradients

• a region between the surface & the freestream whose thickness increases in the flow direction

• manifested by a surface shear stress (Tau_s) that provides a drag force (F_D)

Thermal (Energy) Boundary Layer

• a consequence of heat transfer between the surface and fluid

• a region of the flow characterized by temp gradients and heat fluxes

• a region between the surface and the freestream whose thickness increases in the flow direction

What is determined by the temp profile (i.e thermal boundary layer)>

• the surface heat flux q’’_s & convection heat transfer coefficient h

If the boundary layers are thick then

the value of h is lower

high K =

high h

The concentration boundary layer

• a consequence of evaporation or sublimation of species A from a liquid or solid surface across which a 2nd fluid species B is flowing

• A region of the flow characterized by species fluxes and concentration gradients

• a region between the surface and freestream whose thickness increases in the flow direction

  • manifested by a surface species flux (N’’_A,s) and a convection mass transfer coeffiencet (h_m)

q’’_s changes as

a function of location on the surface, so h does too

the local fiction coefficent (C_f) depends only on

location and Re # for a given geometry

Dimensional analysis can only tell us

the min # of variables we need to worry about to determine Nu (or h) for a given surface geometry and flow

Forced Convection (External flows)

• flat plates

• cylinder in cross-flow

• non-cylinder in cross flow

• flow over a sphere

• flow over banks of tubes

• jets and arrays of jets impinging on plates

  • round nozzles, slot nozzles

• flow through packed beds of solid particles

Forced Convection (Internal flows)

• Circular tubes

  • fully-developed

  • laminar or turbulant

• Non-circular tubes & annuli

• micro-channel flow, nano-chan

Hydrodynamic effects

Assume laminar flow w/ uniform velocity profile at inlet of a circular tube

where does the velocity boundary layer develope?

on the surface of the tube and thickens with increasing x

does the centerline velocity change w/ increasing x?

eventually, the velocity profile becomes parabolic & independent of x

the flow is then hydrodynamically fully develoepd

where does the thermal boundary layer develop

on the surface of the tube and it thickens with increasing x

at the inlet of a circular tube w/ either a uniform surface temp or a constant heat flux through the surface

assume laminar flow with uniform temp T(r,0)=T_i

how does the temp profile depend on x in the fully developed region

it doesn’t

mean velocity is defined in

terms of mass flow rate

for incompressible flow

mass-flow-averaged velocity is equivalent to area-averaged velocity

mean temp is defined by

the thermal energy transported w/the mass flow through a cross section

entry lengths depend on 

the growth rate of the boundary layers, thus, depends on whether the flow is laminar or turbulent, which, in turn, depends on Re #

onset of turbulence occurs at

a critical Re # of Re_D,c = 2300 for internal pipe flow (laminar

fully turbulent conditions exist for

Re_D = 10,000

The local Nusselt # is constant throughout the 

fully developed region, but its values depends on the surface thermal condition

What is Nu used for?

a fully-developed, laminar flow in a pipe but it depends on the boundary conditions

greater mass flow =

change in temp is larger (T_m increases)

• delta T = T_s - T_m

when determining Nu_d average for any tube geometry or flow condition, properties are typically evaluated at

the mean, mean temp

the local Nusselt #  in laminar flow in a circular tube is constant throughout

the fully developed region, but tis value depends on the surface thermal conditions