# Thermal

Written by Jerry Ratzlaff on . Posted in Thermodynamics

## Thermal Conductivity

Thermal conductivity is the ability to transfer heat within a material without any motion of the material.  Depending on the material, the transfer rate will vary.  The lower the conductivity, the slower the transfer.  The higher the conductivity, the faster the transfer.

### Thermal Conductivity formula

$$k = \frac{Ql}{A \Delta T}$$          $$thermal \; conductivity \;=\; \frac { amount \; of \; heat \; transfer \; through \; a \; material \;\;x\;\; length \; or \; thickness \; of \; material } { area \; of \; the \; object \;\;x\;\; temperature \; differential }$$

$$k = \alpha \rho Q$$          $$thermal \; conductivity \;=\; thermal \; diffusity \;\;x\;\; density \;\;x\;\; amount \; of \; heat \; transfer \; through \; a \; material$$

Where:

$$k$$ or $$\lambda$$ (Greek symbol lambda) = thermal conductivity

$$Q$$ = amount of heat transfer through a material

$$l$$ = length or thickness of material

$$A$$ = area of the object

$$\Delta T$$ = temperature differential

$$\alpha$$ (Greek symbol alpha) = thermal diffusity (heat transfer rate)

$$\rho$$ (Greek symbol rho) = density

$$Q$$ = specific heat capacity

## Thermal Conductivity Constant

### Thermal Conductivity Constant formula

$$k_t = \frac {\dot {Q}_t l} {\Delta T}$$          $$thermal \; conductivity \; constant \;=\; \frac { heat \; transfer \; rate \;\;x\;\; length } { temperature \; differential }$$

Where:

$$k_t$$ = thermal conductivity constant

$$\dot {Q}_t$$ = heat transfer rate

$$l$$ = length

$$\Delta T$$ = temperature differential

Solve for:

$$l = k_t \frac {\Delta T} {\dot {Q}_t}$$

## Thermal Diffusivity

Thermal diffusivity ( $$a$$ ) is a measure of the transient thermal reaction of a material to a change in temperature.

### Thermal Diffusivity formula

$$\alpha = \frac {k_t} {p Q }$$          $$thermal \; diffusity \;=\; \frac { thermal \; conductivity \; constant } { density \;\;x\;\; specific \; heat \; capacity }$$

Where:

$$\alpha$$ (Greek symbol alpha) = thermal diffusity (heat transfer rate)

$$k_t$$ = thermal conductivity constant

$$p$$ = density

$$Q$$ = specific heat capacity

Solve for:

$$k = \alpha p Q$$

$$p = \frac {k} {\alpha Q}$$

$$Q = \frac {k} {\alpha p}$$

## Thermal Energy

Thermal energy ( $$Q$$ or $$TE$$ ) (also called heat energy and heat transfer) is the exertion of power that is created by heat, or the increase in temperature.

### Thermal Energy formula

$$Q = mc \Delta T$$          $$thermal \; energy \;=\; mass \;\;x\;\; speciric \; heat \;\;x\;\; temperature \; differential$$

Where:

$$Q$$ or $$TE$$ = thermal energy

$$m$$ = mass

$$c$$ = specific heat

$$\Delta T$$ = temperature differential

Solve for:

$$c = \frac {Q } {m \Delta T}$$

$$\Delta T = \frac {Q}{mc}$$

## Thermal Expansion

The increase in length, area or volume due to the increase (in some cased decrease) in temperature.  The stored energy in the molecular bonds between atoms changes when the heat transfer occurs.  The length of the molecular bond increases as the stored energy increases.

Area thermal expansion - Expands twice as much as lengths do.

Linear thermal expansion - can only be measured in the solid state. The expansion is proportional to temperature change.

Volumetric thermal expansion - can be measured for all substances (liquid or solid) of condensed matter.  Expands three times as much as lengths do.

Some substances such as water can increase or decrease depending on the temperature.

### Thermal Expansion Gases Formula

$$pV = n R T$$ = ideal gas law          $$pressure \; volume \;=\; number of moles of gas \;\;x\;\; specific \; gas \; constant \;\;x\;\; temperature$$

### Thermal Expansion Liquids Formula

$$\Delta V = \beta V_o \Delta T$$ = volumetric or cubical expansion          $$volume \; differential \;=\; volumetric \; thermal \; expansion \; coefficient \;\;x\;\; origional \; volume \; of \; object \;\;x\;\; temperature \; differential$$

### Thermal Expansion Solids Formulas

$$\Delta l = \alpha l_o \Delta T$$ = linear expansion                              $$length \; differential \;=\; linear \; thermal \; expansion \; coefficient \;\;x\;\; initial \; length \; of \; object \;\;x\;\; temperature \; differential$$

$$\Delta A = 2 \alpha A_o \Delta T$$ = aerial or superficial expansion          $$area \; differential \;=\; 2 \;\;x\;\; linear \; thermal \; expansion \; coefficient \;\;x\;\; origional \; area \; of \; object \;\;x\;\; temperature \; differential$$

$$\Delta V = 3 \alpha V_o \Delta T$$ = volumetric or cubical expansion          $$volume \; differential \;=\; 3 \;\;x\;\; linear \; thermal \; expansion \; coefficient \;\;x\;\; origional \; volume \; of \; object \;\;x\;\; temperature \; differential$$

### Formula Definations

$$p$$ = pressure

$$V$$ = volume

$$n$$ = number of moles of gas

$$R$$ = specific gas constant (gas constant)

$$T$$ = temperature

$$\alpha$$ (Greek symbol alpha) = linear thermal expansion coefficient

$$\beta$$ (Greek symbol beta) = volumetric thermal expansion coefficient

$$V$$ = volume of the object

$$\Delta A$$ = area differential

$$\Delta l$$ = length differential

$$\Delta V$$ = volume differential

$$A_o$$ = origional area of object

$$l_o$$ = initial length of object

$$V_o$$ = origional volume of object

$$\Delta T$$ = temperature differential

## Area Thermal Expansion

Area thermal expansion (also known as aerial thermal expansion) happens when any change in temperature expands the area.

### Area Thermal Expansion FORMULA

$$\Delta A = \gamma A_o \Delta T$$          $$area \; differential \;=\; area \; thermal \; expansion \; coefficient \;\;x\;\; initial \; area \; of \; object \;\;x\;\; temperature \; differential$$

$$\frac { \Delta A } { A_o } = \gamma \Delta T$$

Where:

$$\Delta A$$ = area differential

$$\gamma$$ (Greek symbol gamma) = area thermal expansion coefficient

$$A_o$$ = origional area of object

$$\Delta T$$ = temperature differential

## Area Thermal Expansion Coefficient

Area thermal expansion coefficient ( $$\gamma$$ ) (also known as coefficient of aerial thermal expansion) is the ratio of the change in size of a material to its change in temperature.

### Area Thermal Expansion Coefficient FORMULA

$$\gamma = \frac { 1 }{ A } \frac {\Delta A } {\Delta T}$$          $$area \;thermal \; expansion \; coefficient \;=\; \frac { 1 }{ area \; of \; object } \;\;x\;\; \frac { area \; differential } { temperature \; differential }$$

Where:

$$\gamma$$ (Greek symbol gamma) = area thermal expansion coefficient

$$A$$ = area of the object

$$\Delta A$$ = area differential

$$\Delta T$$ = temperature differential

## Linear Thermal Expansion

Linear thermal expansion (also known as line thermal expansion) is a porportional change in the origional length and change in temperature due to the heating or cooling of an object.

### Linear Thermal Expansion formula

$$\Delta l = \alpha l_o \Delta T$$          $$length \; differential \;=\; linear \; thermal \; expansion \; coefficient \;\;x\;\; initial \; length \; of \; object \;\;x\;\; temperature \; differential$$

$$\frac { \Delta l } { l_o } = \alpha \Delta T$$

Where:

$$\Delta l$$ = length differential

$$\alpha$$ (Greek symbol alpha) = linear thermal expansion coefficient

$$l_o$$ = initial length of object

$$\Delta T$$ = temperature differential

## Linear Thermal Expansion Coefficient

Linear thermal expansion coefficient ( $$\alpha$$ ) (also known as coefficient of linear thermal expansion) is the ratio of the change in size of a material to its change in temperature.

### Linear Thermal Expansion Coefficient FORMULA

$$\alpha_l = \frac { 1 }{ l } \frac {\Delta l } {\Delta T}$$          $$linear \;thermal \; expansion \; coefficient \;=\; \frac { 1 }{ length \; of \; object } \;\;x\;\; \frac { length \; differential } { temperature \; differential }$$

Where:

$$\alpha$$ (Greek symbol alpha) = linear thermal expansion coefficient

$$l$$ = length of the object

$$\Delta l$$ = Length differential

$$\Delta T$$ = temperature differential

## Volumetric Thermal Expansion

Volumetric thermal expansion (also known as volume thermal expansion) takes place in gasses and liquids when a change in temperature, volume or type of substance occures.

### Volumetric Thermal Expansion FORMULA

$$\Delta V = \beta V_o \Delta T$$          $$volume \; differential \;=\; volumetric \; thermal \; expansion \; coefficient \;\;x\;\; initial \; volume \; of \; object \;\;x\;\; temperature \; differential$$

$$\frac { \Delta V} { V_o } = \beta \Delta T$$

Where:

$$\beta$$ (Greek symbol beta) = volumetric thermal expansion coefficient

$$\Delta V$$ = volume differential

$$V_o$$ = origional volume of object

$$\Delta T$$ = temperature differential

## volumetric Thermal Expansion Coefficient

Volumetric thermal expansion coefficient ( $$\beta$$ ) (also known as coefficient of volumetric thermal expansion) is the ratio of the change in size of a material to its change in temperature.

### Volumetric Thermal Expansion Coefficient FORMULA

$$\beta = \frac { 1 }{ V } \frac {\Delta A } {\Delta T}$$          $$volumetric \;thermal \; expansion \; coefficient \;=\; \frac { 1 }{ volume \; of \; object } \;\;x\;\; \frac { area \; differential } { temperature \; differential }$$

Where:

$$\beta$$ (Greek symbol beta) = volumetric thermal expansion coefficient

$$V$$ = volume of the object

$$\Delta V$$ = area differential

$$\Delta T$$ = temperature differential