A Primer In Heat Transfer

[Crazy_Nate]

This is a sort of teaching medium, courtesy of me...

 

Why me? Well, I have my BS in Mechanical Engineering, from RPI. I did my concentration in Energy Systems. I am currently in my second and final year of my MSME. I enjoy teaching.

 

Why here? Certain understandings of all things thermal have been demonstrated to be in error, so, I will do my best to disseminate some of the knowledge that I've obtained throughout my studies to the OCC community.

 

So, what is this? It is an attempt of mine to answer some of the basic and in depth questions about the general application of the mechanisms and background information on the topic of heat transfer.

 

 

So, with all that, have a little patience and I'll put everything up as I get to it. Housekeeping first .

 

 

I'll edit this post later to explain some of my notation, or clarify general things about the thread.

 

Notation notes:

• Greek symbols are spelled out
  
• The underscore character denotes a subscript (x_y)
  
• The caret character denotes a superscript, and in most cases a power (x^y)
  
• Subscripts that are "_dot" serve the same purpose as putting a dot on top of the preceding symbol. Aka W_dot = P
  

[Crazy_Nate]

Intro.

 

Heat transfer is a discipline that deals with the transfer of energy, specifically, thermal energy.

 

Calculus (specifically differentiation, integration), physics (Newton's laws of motion, conservation of momentum, energy), and thermodynamics are preludes to this topic. I will attempt to put things in laymen's terms whenever possible, however, some topics can be a little too complicated.

 

We deal with many forms of heat transfer in our daily lives. Turning on a stove burner transfers heat to something place on top of it, as well as the surroundings. But, what exactly is happening?

 

Heat and Energy.

 

Energy exists in many forms: Mechanical, thermal, kinetic, potential, electrical, magnetic, chemical, and nuclear. The sum of which add up to be the total energy (E) of the system or matter of which you are concerned. The energy associated with the molecular structure, the "internal stuff" adds up to be the internal energy (U). It's a kind of sum of all the kinetic and potential energies of all the particles / molecules. The amounts of energy associated with a phase of the system (such as solid, liquid or gas) are latent energy / heat.

 

Specific Heat.

 

Specific heat is defined as the amount of energy necessary to raise the temperature of a unit of mass by one degree. Engineers usually use two types of specific heats, c_p and c_v, the difference is the type of process for which each is appropriate, c_p for constant pressure processes and c_v for constant volume processes.

 

Example: Delta U = m*c_p*Delta T for a constant pressure process.

 

Energy Transfer.

 

Energy can be transferred to a mass through heat transfer or work. Heat transfer is driven by a temperature difference. Power is the time derivative of (time rate of change of...) work denoted P or W_dot.

 

First Law of Thermodynamics.

 

The first law of thermodynamics is written in a couple different ways. It involves a balance of energy, aka "conservation of energy." Q - W = Delta U. The minus is a sign convention. It can also be written E_in - E_out = Delta U...which gives you a little insight to the previous equation -- Heat in is positive, work out (aka work done by the system) is positive.

Steady state systems.

 

Using certain quantities like mass flow rate (mass per unit time - m_dot), density (mass per unit volume - rho), velocity (length per unit time - V), and cross sectional area (length^2 - A_c) we can write some of the previous equations to allow for changing quantities.

 

m_dot = rho*V*A_c (definition, useful equation, can be shown with a simple unit analysis)

 

Q_dot = m_dot*c_p*Delta T, etc.

Heat transfer: Conduction

 

Conduction involves the transfer of kinetic energy from a more energetic particle to a neighboring - less energetic - particle. It happens in solids, liquids and gases.

 

The rate of conduction through a material depends on the thickness of the material, the thermal conductivity of the material, and the temperature gradient.

 

Q_dot_cond = - k*A*Delta T / Delta x

 

Note: Delta x is the thickness of the material in the direction of the heat transfer (the x-direction, specifically). The negative sign exists because heat ALWAYS flows from hot to cold (without the addition of work input, such as a refrigeration cycle...).

 

The equation stated above in the limiting case where Delta x approaches zero becomes Fourier's law of heat conduction:

 

Q_dot_cond = -k*A*dT/dx

 

Where dT/dx is the temperature gradient, or the spatial derivative of temperature with respect to the x-direction.

 

Thermal Conductivity.

 

Thermal conductivity is defined by Fourier's law. In words, this is the rate of heat transfer through a unit thickness of a material per unit area per unit temperature difference. Materials with higher conductivities will conduct heat more readily than those of lower conductivity all else held constant. Thermal conductivities are not constant at all temperatures, but sometimes they are approximated as such.

 

Heat transfer: Convection.

 

Convection involves the transfer of heat from a surface to a fluid (gas or liquid) that is in contact with it. It combines the affects of fluid motion and conduction. As you can imagine, the faster the fluid is moving, the more readily the heat transfers.

 

Convection falls into two categories: Natural and forced. Natural convection involves fluid motion that is driven by buoyancy (basically, density differences cause fluid motion - think of hot air rising). Forced convection involves some external force causing fluid motion, like a fan or pump. In reality, most convection involves a combination of both (called mixed convection). However, to simplify, for certain processes one or the other may be negligible.

 

There are other more complicated convection processes that involve phase changes, such as boiling heat transfer or condensation.

 

Even with all the different types of convection, all of the heat transfer may be described with one equation, Newton's law of cooling:

 

Q_dot_conv = h*A_s*(T_s - T_infinity)

 

Where h is the heat transfer coefficient, A_s is the surface area, T_s is the surface temperature, and T_infinity is the fluid free stream temperature, sufficiently far away from the surface (there exists what is called a boundary layer between the surface and the fluid far away, in which the temperature varies from the surface temperature to the free stream temperature).

 

Several things are noticeable when you look at the equation. Increase the heat transfer coefficient, increase heat transfer. Increase surface area, increase heat transfer. Decrease temperature difference, decrease heat transfer (the closer the surface temperature goes to the ambient temperature, the less readily heat will transfer...with zero heat transfer occurring at no temperature difference). Note: One can see why it is impossible to cool a computer component beyond ambient temperatures with heat transfer from the ambient air.

Heat Transfer: Radiation.

 

Radiation is energy emitted by electromagnetic waves. It is different from conduction and convection in that it does not need a medium (solid, liquid or gas) to occur.

 

The Stefan-Boltzmann law describes the maximum rate at which a surface can radiate in idealized conditions (an ideal surface known as a black body):

 

Q_dot_emit_max = sigma*A_s*T_s^4

 

Where, sigma is the Stefan-Boltzmann constant, A_s is the surface area, T_s is the absolute temperature (K or R).

 

Real bodies are less than ideal, and the radiation is less:

 

Q_dot_emit = epsilon*sigma*A_s*T_s^4

 

Where epsilon is the emissivity of the surface, a property of the surface (between 0 and 1).

 

There are related equations for incident radiation.

 

Q_dot_incident = Q_dot_absorbed + Q_dot_reflected

Q_dot_absorbed = alpha*Q_dot_incident

Q_dot_reflected = (1 - alpha)*Q_dot_incident

 

Where alpha is the absorptivity of the surface.

 

Often, radiation is expressed between one surface and another 'surrounding' surface:

 

Q_dot_radiation = epsilon*sigma*A_s*(T_s^4 - T_surrounding^4)

 

Sometimes it is easier to describe total heat transfer by lumping radiation into Newton's law of cooling, by using a 'combined heat transfer coefficient':

 

Q_dot_total = h_combined*A_s*(T_s - T_infinity)

[Crazy_Nate]

Heat Conduction Equation

Internal Heat Generation

Spherical, cylindrical, rectangular forms

Steady state

Boundary conditions

 

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[Crazy_Nate]

Steady conduction

Thermal resistances, "circuits"

Contact resistances

Fins!

Special conditions - three cases for fins.

Fin efficiency, fin effectiveness

 

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[Crazy_Nate]

Transient conduction

Basic description

Additional terms and complications

 

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[Crazy_Nate]

Convection!

No slip conditions

Boundary layers

Nusselt number

 

Flow:

Viscous / inviscid

Internal / external

Compressible / incompressible

Laminar / transitional / turbulent

Natural / mixed / forced

Steady / unsteady

 

Velocity boundary layer / shear stress

Thermal BL / Prandtl number

 

Laminar / turbulent <= Reynolds number

 

Continuity / momentum / cons. of energy equations.

 

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[Crazy_Nate]

Special topics

 

Natural convection

Boiling and condensation

 

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[Crazy_Nate]

Devices:

Heat exchangers

Heat pipes

Etc.

 

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[Crazy_Nate]

Thermodynamic cycles, briefly!

 

Think phase change cooling

 

 

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[Crazy_Nate]

Other stuff here!

 

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[dr_bowtie]

Wow...Nate....

 

it looks like you plan to be very thorough in your explanation....

 

I couldnt get myself to read more than the first paragraph.

 

I think what would help alot is a simple easy statement of what type of cooler would work best for air cooling and why...?

 

 

you know there has been much debate over copper and aluminum heatsink and arguments on which is better and why....

 

someone needs to set the record straight...if a copper inner workings with aluminum fins for heat dissipation is better than just one metal....?

 

I think your detailed elaboration may better help those that are interested in alternative cooling systems....IE: chilled water and phase or even cascade cooling....you need more physics there than you do simple air coolers...

 

other than that great work...

[Crazy_Nate]

Wow...Nate....

 

it looks like you plan to be very thorough in your explanation....

 

I couldnt get myself to read more than the first paragraph.

 

I think what would help alot is a simple easy statement of what type of cooler would work best for air cooling and why...?

 

 

you know there has been much debate over copper and aluminum heatsink and arguments on which is better and why....

 

someone needs to set the record straight...if a copper inner workings with aluminum fins for heat dissipation is better than just one metal....?

 

I think your detailed elaboration may better help those that are interested in alternative cooling systems....IE: chilled water and phase or even cascade cooling....you need more physics there than you do simple air coolers...

 

other than that great work...

 

Hey doc, just getting started...I'm planning on hitting this topic with 'freight train' type proportions.

 

Hopefully, I'll get around to adding tons of links today, with special attention to some of the background stuff. Basic physics links, some calculus, explain my notation a little better, etc etc.

 

I'm trying not to make this too complicated, but touch on many of the topics that help describe these things (relevant to computers, especially)...but, sometimes it's hard toning it down without it losing some meaning.

 

I'll get some more stuff up, and people can let me know where they get stuck in the understanding, and I'll work on it a little more. It's going to be a little bit of an ongoing process for me.

 

Thanks for the input!

 

PS...I'll get to more of the computer specific stuff after I've hammered out some of the basics. As with a lot of engineering, "better" can be subjective, as function is not the only design criteria. I'll try to get this in detail, to make it nice and clear.