Heat Transfer Final

Heat Transfer/Heat

Thermal energy in transit due to a spatial temperature difference

Conduction

A mode of heat transfer involving random microscopic molecular motion (diffusion)

Convection

A mode of heat transfer involving diffusion + bulk motion of the fluid at a microscopic level

Radiation

A mode of heat transfer involving electromagnetic waves and does not require a medium

What mode of heat transfer is Fourier’s Law used for?

Conduction

Variable Q and its units

Heat, J

q, Qdot, Q” and its units

Heat transfer rate, W or J/s

What mode of heat transfer is Newton’s Law of Cooling used for?

Convection

variable h and its units

Convection heat transfer coefficient, W/m²K

What mode of heat transfer uses the Stefan-Boltzmann Law?

Radiation

What equation does a black body use to calculate heat transfer?

Stefan-Boltzmann Law

variable e and its units

emissivity, unitless, will always be between 0 and 1

Value of Stefan-Boltzmann constant

5.67×10^-8

Radiation intensity formula

E/pi

Variable I and its units

Radiation intensity, W/m²(sr)

Intensity

Emissive power per solid angle. Intensity is the same everywhere in space if we are considering the same solid angle

For a diffuse surface, is intensity independent of direction?

Yes, intensity is independent of direction

What is emitted radiation a function of?

Wavelength

What two relations does Wien’s Displacement Law prove for a blackbody?

1. As temperature increases, emissive power increases

2. As temperature increases, radiation appears in shorter wave length

What range of temperature is in the infrared region, making it out of the visible region?

T<800K

variable G and its units

Incident radiation, w/m²(um)

Does the incident radiation or emissive power use the area of the receiving object?

Incident radiation

Does the incident radiation or emissive power use the area of the emitting object?

Emissive power

Black body definition

1. Absorbs all incident radiation, regardless of wavelength and direction

2. For a prescribed temperature and wavelength, no surface can emit more energy than a blackbody

3. Although emitted radiation by a blackbody is a function of wavelength and temperature, it is independent of direction. A blackbody is a diffuse emitter.

Gray surface

Absorptivity and emissivity are independent of wavelength

In what 2 cases are emissivity and absorptivity equal?

1. Irradiation G=E_b and E_wavelength,b=G_wavelength

2. Gray surface

When is transmissivity equal to zero?

Opaque surface

F₁₂

Fraction of radiation that travels from object 1 to 2

Reciprocity Relation

A₁F₁₃=A₃F₃₁

Conduction Heat Transfer/ Heat Transfer by Diffusion

Transport of energy in a medium due to a temperature gradient, and the physical mechanism is that of random atomic or molecular activity

Which variable contributes to the temperature increase or decrease of the object in the heat diffusion equation?

E_stored

What are the four possible boundary conditions for thermal diffusion?

1. Constant surface temperature

2. Constant surface heat flux

3. Adiabatic or insulated surface

4. Convection surface condition

Unless otherwise stated, what 3 assumptions should you make for temperature diffusion problems?

1. Steady State

2. 1 Dimensional

3. Constant properties

Cases of thermal resistance problems

1. Composite wall

2. Contact Resistance

3. Radial System

2 types of resistances

1. Parallel

2. Series

Fins

Extended heat transfer surfaces

4 cases for fin conduction problems

1. Convection from the tip

2. Adiabatic Tip (convective heat loss from tip is negligible)

3. Tip with constant temperature

4. Very long fin

What case of fin conduction would you use for “a very long rod”

Very long fin

What type of problem utilizes a modified version of the Bessel equations?

Fins of non-uniform cross-sectional area

What happens to the space between fins when you increase the number of fins?

The space decreases

Separation of Variables

Assume the final form of the solution can be separated to the product of two functions

What type of problem utilizes separation of variables?

2-D Steady State Conduction

variable S

Shape factor

variable q^*_ss

Dimensionless conduction heat rate

When do you use the lumped method?

0-Dimensional Transient Conduction

What is B_i

Biot Number

What is F₀

Fourier Number

What happens if B_i is high?

k, thermal conductivity, is low

What should you assume in quenching problems unless otherwise stated?

1. Transient

2. 0D Lumped (check if valid)

3. Constant properties

When is lumped approach valid?

B_i<0.1

Is the cylindrical coordinate system used for solid cylinders or hollowed cylinders?

Solid cylinders

3 cases of 1D Transient Conduction w/ a Semi-Infinite Solid

1. Constant Surface Temperature

2. Constant Surface Heat Flux

3. Surface Convection

What mathematical approaches can be used for finding node temperatures under steady state conduction?

1. Successive substitution

2. Matrix Inversion

3. Gauss-Seidel Method

When should you use transient conduction methods for 1D conduction?

Time is involved in the problem

No Slip Condition

Velocity is zero at the surface

Viscous Flow

Flow in which frictional effects are significant

Internal Flow

Flow inside a pipe or duct

External Flow

Flow over a plate, cylinder, sphere, etc.

Incompressible Flow

Density remains nearly constant

Are liquids compressible or incompressible?

Incompressible

When can gas flows be approximated as incompressible?

The density change is under 5%

What are the 3 governing equations for flow?

1. Continuity

2. Momentum

3. Energy

What is the film temperature for external flows?

The average between surface and fluid temperature

What cases should use external flow calculations?

1. Flat plate (laminar or turbulent)

2. Cylinder

3. Sphere

4. Bank of Tubes

5. Impinging Jet

What is a mixed boundary layer condition?

First half laminar, second half turbulent

When does A=0?

Entire flat plate is assumed to have turbulent flow, even from the leading edge

When does A=871?

Flat plate has mixed boundary condition

When should you assume isothermal?

Mixed boundary conditions & unheated starting length

When should you assume non-isothermal?

Flat plate with constant heat flux

When should you use the Hilpert relation?

1. Single, isolated cylinder in cross-flow

2. Moderate Reynolds numbers (10<Re<10⁵)

3. Desiring a simple correlation with tabulated constants (C, m)

When shoud you use Jakob relation?

1. Flow over a single cylinder in cross-flow

2. Lower Reynolds number ranges (laminar or early transitional regimes)

When can Hilpert’s correlation be used for non circular shapes?

Using the characteristic length as the baseline property or a specific C and m are provided in the problem statement

When is Zhukauskas correlation used?

1. Tube bundles

2. External cross-flow over in-line or staggered tube arrays

3. Need row-wise correction factors, bundle correction factors, and geometry-based adjustments

4. Heat exchanger tube banks

When is Churchill & Bernstein correlation used?

1. Single cylinder in cross-flow

2. Flow is in very wide range of Reynolds and Prandtl numbers

3. Desire more general correlation

When is the Ranz & Marshall correlation used?

1. Small particles, droplets, or bubbles

2. Single spherical particle in a fluid

3. Low-to-moderate Reynolds numbers

4. Forced or natural convection

5. Particle diameter is small enough that property variations around the particle are negligible

What should you assume if using Ranz & Marshall?

1. Nearly isothermal surface

2. Minimal temperature dependence of properties

When is the Whitaker correlation used?

1. Flow over a sphere

2. Wide Reynolds number range

3. Property variations between surface and bulk

4. Desire better accuracy for large temperature differences

5. Particle/sphere is rigid

What correlation should be used for flows over a sphere when Re<200?

Ranz & Marshall

What correlation should be used if there is a high temperature difference or property variation?

Whitaker

For external flow over a bank of tubes, where are all properties evaluated at?

Film temperature

What should you do for internal flow if T_o is not provided?

Make a reasonable assumption for property table readings

Where are properties evaluated at for internal flow?

Mean temperature

What do you need to calculate the heat transfer rate per unit length of a rectangular channel?

h and an entrance condition

Types of heat exchangers

1. Fin & Tube

2. Shell & Tubes

3. Plate Heat Exchangers

4. Microchannel

5. Concentric Tube

Fin and Tube Heat Exchanger

Mostly used for liquid to air applications or two phase flow. With fins, air is unmixed in a transverse direction. Without fins, the air flow is mixed.

Shell and Tubes Heat Exchanger

Mostly used for liquid to liquid applications or two phase flow

Plate Heat Exchangers

Liquid to liquid or two phase flow

Microchannel Heat Exchanger

Liquid to gas applications or two phase flow

Concentric Tube Heat Exchanger

Liquid to liquid applications or two phase flow

Types of flow

1. Parallel

2. Counter