Thermal and Thermo-mechanical Design

FUNCTION AND CONSTRAINTS

MAXIMIZE1

MINIMIZE1

Thermal insulation, transitional

 thermal insulation, transitional optimize thermal stability; temperature difference, time fixed; thickness free

1 / a = ρ Cp / λ

a = λ / ρ Cp

Thermal insulation, cyclic heating

 thermal insulation, cyclic heating optimize energy consumption per cycle; temperature rise, cycle time fixed; wall thickness free

(√a) / λ =
(λρCp)-1/2

λ / (√a) =
(λρCp)1/2
optimize mass; temperature rise, cycle time, wall thickness fixed

(√a) / λ =
((ρCp)/λ)1/2

λ / (√a) =
(λ/(ρCp))1/2

Thermal insulation, steady state

 thermal insulation, steady state optimize mass; minimum heat flux at steady state; thickness specified

ρ / λ

λ / ρ

Heat exchanger tubes

 heat exchanger maximum heat flux per unit area; no failure under Δp

λ σy

1 / λ σy
optimize mass; maximum heat flux per unit mass; pressure and temperature difference, tube radius fixed; wall thickness free

σy2λ / ρ

ρ / σy2λ

Short-term thermal energy storage

 short term thermal energy storage optimize energy stored for given temperature rise and time

λ / (√a) =
(λρCp)1/2

(√a) / λ =
1 / (λρCp)1/2
maximize energy stored for given temperature rise and time

1 / (ρ√a) =
(Cp/λρ)1/2

ρ√a =
(λρ/Cp)1/2

Long-term thermal energy storage

 long term thermal energy storage maximum energy stored / unit material cost (storage heaters)

Cp / Cm

Cm / Cp

Thermal stability

 thermal stability minimize thermal distortion for given heat flux

λ / α

α / λ

Thermal shock resistance

 thermal shock resistance maximum change in surface temperature; no failure

σy / Eα

E α / σy

Heat sinks

   maximum heat flux per unit volume; expansion limited

λ / Δα

Δα / λ
maximum heat flux per unit mass; expansion limited

λ / ρ Δα

ρ Δα / λ

  1. λ = thermal conductivity;
    a = thermal diffusivity;
    Cp = specific heat capacity;
    Cm = material price ($/kg*)
    (Cm ρ = material price per unit volume)
    α = thermal expansion coefficient;
    E = Young's modulus;
    ρ = density;
    σy = failure strength (the yield strength for metals and ductile polymers, the tensile strength for ceramics, glasses and brittle polymers);
    Full list of symbols

    * Units can be changed in CES EduPack under Settings.