Temperatures in expressions

Temperature units in expressions

The CAE expression system allows for temperatures in Celsius, Kelvin, Fahrenheit, and Rankine. When specifying temperature units in the formula for an expression, you must designate whether the temperature unit represents a temperature in the context of thermodynamic equilibrium or a temperature difference.

Note:

In the expression system, temperature in the context of thermodynamic equilibrium is referred to as temperature.

Temperature scale	Temperature unit	Temperature difference unit
Celsius	C	dC
Kelvin	K	dK
Fahrenheit	F	dF
Rankine	R	dR

Special rules for expressions that contain temperatures and temperature differences

The following table lists the ways you can combine temperature units and temperature difference units. If the combination is invalid, the expression system issues an error. The examples in the table use constants as the arguments for the operations. However, the arguments can also be the temperature variable, the auto-generated expression for the temperature variable, or functions or other expressions that evaluate to the same temperature unit as the constant in the table.

Operation (1)	Result (1)	Examples	Example results in base units (2)
T+T	Error	800[K]+300[K]	Error
800[C]+300[K]
800[C]+300[F]
T–T	dT	800[K]–260[K]	540[dC]
800[C]–260[K]	813[dC]
800[C]–260[F]	673[dC]
T+dT	T	800[K]+360[dK]	887[C]
800[C]+360[dK]	1160[C]
800[C]+360[dF]	1000[C]
T–dT	T	800[K]–360[dK]	167[C]
800[C]–360[dK]	440[C]
800[C]–360[dF]	600[C]
dT+T	T	800[dK]+360[K]	887[C]
800[dC]+360[K]	887[C]
800[dC]+360[F]	983[C]
dT–T	Error	800[dK]–360[K]	Error
800[dC]–360[K]
800[dC]–360[F]
dT+dT	dT	800[dK]+360[dK]	1160[dC]
800[dC]+360[dK]	1160[dC]
800[dC]+360[dF]	1000[dC]
dT–dT	dT	800[dK]–360[dK]	440[dC]
800[dC]–360[dK]	440[dC]
800[dC]–360[dF]	600[dC]
T∙T	Error	360[K]*18[K]	Error
360[C]*18[K]
360[C]*18[F]
T∙dT	Error	360[K]*18[dK]	Error
360[C]*18[dK]
360[C]*18[dF]
T/T	Error	360[K]/18[K]	Error
360[C]/18[K]
360[C]/18[F]
T/dT	Error	360[K]/18[dK]	Error
360[C]/18[dK]
360[C]/18[dF]
dT∙dT	Error	360[dK]*18[dK]	Error
360[dC]*18[dK]
360[dC]*18[dF]
dT∙T	Error	360[dK]*18[K]	Error
360[dC]*18[K]
360[dC]*18[F]
dT/dT	Dimensionless	360[dK]/18[dK]	20
360[dC]/18[dK]	20
360[dC]/18[dF]	36
dT/T	Error	360[dK]/18[K]	Error
360[dC]/18[K]
360[dC]/18[F]
(1) T represents a quantity that has temperature as the unit. dT represents a quantity that has temperature difference as the unit.(2) The software uses the exact conversions for temperature units. However, for simplicity, the following approximate conversions are used to calculate the example results: 0 K = –273 C; 0 R = –460 F.

Using temperatures in the formula for an expression

For most FE applications that involve temperatures, the temperatures are temperature differences.

• In thermal stress problems, the thermal strain is related to temperature change relative to a reference temperature where there is no thermal strain.

• In heat transfer problems, conduction heat transfer is a function of temperature gradient, and convection heat transfer is related to temperature difference.

FE applications that involve temperature in the context of thermodynamic equilibrium are equations of state and radiation heat transfer.

A general strategy for using temperatures in a formula for an expression is to convert all temperatures to temperature differences that are defined relative to 0 K. By doing so, you:

• Avoid the limitations on operations that are inherent with temperatures.

• Allow expressions to be used to model equations of state and radiation heat transfer.

Modeling an equation of state

The density of air at atmospheric pressure as a function of temperature for temperatures near room temperature is:

ρ = 101 kPa / ((0.287 kJ / kg-K) T)

where T is absolute temperature in Kelvin.

To create an expression for this relationship, select a unit of mass density for the expression. For the formula for the expression, type:

101[mN/mm^2(kPa)]/(0.287[kJ/kg-dK]*(ug_var(“temperature”)-0[K]))

Because the difference between two temperatures is a temperature difference, the expression system evaluates (ug_var(“temperature”)–0[K]) to the unit of [dK]. During processing, the expression system cancels the instances of [dK].

Modeling radiant heat transfer

The heat flux between a blackbody at absolute temperature T in Kelvin and a blackbody at 100 K is:

5.669 x10–8 W/m2-K4 (T4 – (100 K)4)

where T is absolute temperature in Kelvin.

However, the expression system does not:

• Allow temperatures or temperature differences to a power

• Recognize the dimensional grouping of the Stefan-Boltzmann constant.

A workaround to this limitation is to:

1. Convert the temperatures to temperature differences.

2. Make the temperature differences dimensionless.

3. Take each dimensionless temperature difference to the fourth power.

4. Assign the dimensional grouping of heat flux to the numerical value of the Stefan-Boltzmann constant.

Thus, to create the formula for the expression, type:

(5.669e-8[W/m^2])*(((ug_var(“temperature”)-0[K])/1[dK])^4–((100[K]-0[K])/1[dK])^4)

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