heat transfer

Law used for conduction

Fourier's law

Law used for convection

Newton's law of cooling

Fourier's law (form)

q = -kA dT/dx (rate) or q'' = -k dT/dx (flux)

At the atomic level, what properties govern the magnitude of thermal conductivity in a solid?

Atomic mass and bond strength

Microscopic-level properties governing thermal conductivity in solids

Specific heat, group velocity, and mean free path

How heat is conducted in metals

Via electrons

At the continuum level, what properties govern thermal conductivity?

density, hardness, purity

Mechanism of heat transfer in nonmetals

phonons (atomic vibrations)

Direction heat moves

Downhill, from hot to cold (opposite the temperature gradient)

The "m" in the units of thermal conductivity (W/mK)

direction of transport

What the heat transfer coefficient depends on

Fluid properties, velocity, and geometry

Mechanism of convection

Bulk motion of atom

T∞

The fluid temperature far enough from the boundary layer that it no longer changes (ambient fluid temperature)

Mechanism of thermal radiation

Spontaneous emission of electromagnetic radiation

What affects emissivity

Material, surface treatment

Units of emissivity

Dimensionless

Allowed values of emissivity

0 to 1

Emissivity is a function of

Wavelength, temperature, and direction (angle)

Emissivity of an ideal emitter (black body)

ε = 1

Why radiation is mathematically challenging

Non-linear in T; depends on wavelength; emissivity depends on surface treatment (not purely a material property)

Tsurr

The temperature of the surfaces that the object can "see"

Difference between T∞ and Tsurr

T∞ is the fluid temperature outside the boundary layer; Tsurr is the surface temperature of the surrounding walls/enclosure

Inverse of thermal resistance

Thermal conductance

Can generation be negative?

Yes, endothermic reaction

Approximations that convert a differential to a difference

One-dimensional, steady state, no generation, constant properties, Cartesian coordinates

Convective resistance

R = 1/(hA)

Pipe radius less than critical radius: how to decrease heat transfer

Add insulation up to at least r_min (the radius where q returns to the bare-pipe value); thin insulation makes it worse

Pipe radius exactly equal to critical radius

Heat transfer is a maximum (resistance is a minimum) with no insulation

When R_planar approximates R_cylinder

When r_o >> r_o - r_i (thin wall relative to radius)

Why q' × pipe length doesn't give total heat loss

The fluid cools as it loses energy, so ΔT is not constant along the pipe

Temperature profile in a plate vs. cylindrical vs. spherical shell

Plate: linear; cylinder: logarithmic (curved); sphere: 1/r (most curved)

Why thermoelectrics are made of "couples"

So that holes (p-type) can carry thermal energy in the same direction as electrons (n-type)

Working fluid in a thermoelectric

Electrons (and holes)

Hole (thermoelectrics)

A positive charge carrier that carries thermal energy

Seebeck coefficient (in words)

How much voltage a material generates per unit temperature difference applied

Units of the Seebeck coefficient (S)

V/K

What sets the operating point on a thermoelectric pump curve

The connection to the environment (thermal load)

Electrical vs. thermal transport in TED couples

Electrical transport is in series; thermal transport is in parallel

Contact resistance when one surface is polished

Decreases

Contact resistance when both surfaces are polished

Decreases (probably)

Contact resistance when a surface is warped

Increases (probably)

Contact resistance when clamped with bolts

Decreases

Contact resistance when the interface is wetted

Decreases

Contact resistance when conductive paste is added

Decreases

Origin of textbook shape factors

Mostly analytic solutions of Laplace's equation; some from experiments

Combining multiple shape factors

Parallel paths: S_tot = Σ S_i (equivalently R_tot = [Σ 1/(kS)_i]⁻¹)

Combining heat transfer from multiple shape-factor components

q_tot = Σ q_i

Effect of doubling burial depth of a long cylinder on S

S decreases, because heat travels over a longer distance so heat transfer decreases (q = kSΔT)

Assumptions required for a shape factor

Steady state, constant properties, no generation (usual assumptions); two isothermal surfaces with every other surface adiabatic (specific to shape factors)

Effect on S of increasing separation between two buried parallel isothermal cylinders

S decreases, because the conduction path lengthens and heat transfer decreases

Dominant forces at low vs. high Reynolds number

Low Re: viscous forces; high Re: inertial (momentum) forces

Average h at the critical Reynolds number

Stays continuous (no jump); its slope changes

Local vs. average Nusselt number in a problem statement

Asks for heat flux means local; asks for heat transfer means average

Exact solution for laminar flat plate flow

Blasius solution

Prandtl's contribution to convection in this class

The boundary layer concept (viscous effects confined to a thin layer near the surface), together with the no-slip condition at the wall

Why the average Nu coefficient is twice the local one (laminar flat plate)

Integrating Re_x^(1/2) ~ x^(1/2) over the length gives a factor of 2

Characteristic length for Reynolds number in internal flow

The hydraulic diameter: Re_D = 4ṁ/(πDμ) for a round tube

What the entrance length depends on

Reynolds number and pipe diameter

Entrance length: turbulent vs. laminar

Turbulent is shorter, because turbulent boundary layers grow faster

Transition location in a long tube

There is none; the flow is either all laminar or all turbulent

Laminar flow in a tube: is h constant?

Yes for fully developed flow, since h = Nu k/D with Nu constant

Mean temperature in a constant-wall-temperature tube

Rises exponentially toward the wall temperature (faster exponential in the entrance region)

Mean temperature in a constant-heat-flux tube

Rises linearly with x once fully developed (with a wall temperature that is parallel to it)

Velocity profile of fully developed laminar flow in a circular tube

Parabolic: v(r) = v_max[1 - (r/R)²]

Maximum velocity vs. mean velocity in laminar tube flow

v_max = 2 v̄ (for v̄ = 0.2 m/s, v_max = 0.4 m/s)

Boundary condition the fin tables use

An insulated (adiabatic) tip; q''(x=L) = -k dT/dx at L = 0

When an insulated tip is justified

When the tip area is small compared with the rest of the fin, or when the fin is long

Two ways to calculate fin heat transfer from the temperature distribution

q = ∫ hP(T - T∞) dx over the length, or q = -kA_c dT/dx evaluated at the base (x = 0)

Effect of doubling fin length on efficiency

Decreases, since q_max doubles but q_fin does not

Theoretical min/max of fin efficiency and effectiveness

Efficiency: 0 to 1; effectiveness: 0 to unbounded (should be greater than 1 for the fin to be worthwhile)

Why use a triangular fin

Weight: it can provide similar heat transfer with less material

Units of β (fin table, section 2.4 of HTref)

1/m

Units of ξ (fin table, section 2.5 of HTref)

Dimensionless

Ways to increase fin efficiency

Make the fin shorter; use a higher-conductivity material

Ways to increase fin effectiveness

Make the fin longer (up to a point); make it thinner; use a higher-conductivity material

Diffusion (heat) equation, constant properties, no generation

V2T=(1/a)dT/dt

Diffusion equation, constant properties, steady state

∇2T + q’’’/k = 0

Diffusion equation, constant properties, 1-D Cartesian

d2T2/dx2 + q’’’/k = (1/a) dT/dt

Convective boundary condition

-kdT/dx |_(x=a) = h(T(a) - Tinf) (non-homogeneous because of the Tinf term)

Control volume for deriving a temperature distribution

Differential control volume

Three kinds of boundary conditions

1st kind: specified temperature (Dirichlet); 2nd kind: specified flux (Neumann); 3rd kind: convection (Robin/mixed)

Integration constants for a 2-D, steady, second-order system

4

Boundary/initial conditions for 2-D transient conduction

5 (4 boundary conditions + 1 initial condition)