Fundamentals of heat transfer Radiative equilibrium Surface properties Non-ideal effects Internal power generation Environmental temperatures Conduction Thermal system components 2002 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Classical Methods of Heat Transfer Convection Heat transferred to cooler surrounding gas, which creates currents to remove hot gas and supply new cool gas Don t (in general) have surrounding gas or gravity for convective currents Conduction Direct heat transfer between touching components Primary heat flow mechanism internal to vehicle Radiation Heat transferred by infrared radiation Only mechanism for dumping heat external to vehicle
Ideal Radiative Heat Transfer Planck s equation gives energy emitted in a specific frequency by a black body as a function of temperature e λb = Stefan-Boltzmann equation integrates Planck s equation over entire spectrum Prad = σt 4 W σ = 567. x10 8 2 4 m K 2 2πhC0 hc λ exp λkt 5 0 1 (Don t worry, we won t actually use this equation for anything ) ( Stefan-Boltzmann Constant )
Thermodynamic Equilibrium First Law of Thermodynamics Q W = du dt heat in -heat out = work done internally Heat in = incident energy absorbed Heat out = radiated energy Work done internally = internal power used (negative work in this sense - adds to total heat in the system)
Radiative Equilibrium Temperature Assume a spherical black body of radius r Heat in due to intercepted solar flux Q = I π r 2 in s Heat out due to radiation (from total surface area) Qout = 4πr σt 2 4 For equilibrium, set equal I πr = 4πr σt I = 4σT s 2 2 4 4 s 1 AU: I s =1394 W/m 2 ; T eq =280 K T eq Is = 4σ 1 4
Effect of Distance on Equilibrium Temp Black Body Equilibrium Temperature ( K) 500 450 400 350 300 250 200 150 100 50 0 0 10 20 30 40 Distance from Sun (AU)
Shape and Radiative Equilibrium A shape absorbs energy only via illuminated faces A shape radiates energy via all surface area Basic assumption made is that black bodies are intrinsically isothermal (perfect and instantaneous conduction of heat internally to all faces)
Effect of Shape on Black Body Temps Black Body Equilibrium Temperature ( K) 700 600 500 400 300 200 100 Spherical Black Body Adiabatic Black Wall Double-Sided Wall 0 0.1 1 10 100 Distance from Sun (AU)
Incident Radiation on Non-Ideal Bodies Kirchkoff s Law for total incident energy flux on solid bodies: QIncident = Qabsorbed + Qreflected + Qtransmitted Q Q Q absorbed reflected transmitted + + =1 QIncident QIncident QIncident Q Q absorbed reflected α ; ρ ; τ Q Q where Incident α =absorptance (or absorptivity) ρ =reflectance (or reflectivity) τ =transmittance (or transmissivity) Incident Q Q transmitted Incident
Non-Ideal Radiative Equilibrium Temp Assume a spherical black body of radius r Heat in due to intercepted solar flux Q = Iαπ r 2 in s Heat out due to radiation (from total surface area) Qout = 4πr εσt 2 4 For equilibrium, set equal (ε = emissivity - efficiency of surface at radiating heat) ε Isαπ r = 4π r εσt Is = 4 T α σ 2 2 4 4 T eq Is = α ε 4σ 1 4
Effect of Surface Coating on Temperature 1 0.9 0.8 Black Ni, Cr, Cu Black Paint 0.7 0.6 0.5 0.4 0.3 Aluminum Paint Teq=560 K (a/e=4) Teq=471 K (a/e=2) Teq=396 K (a/e=1) Teq=333 K (a/e=0.5) Teq=280 K (a/e=0.25) 0.2 Polished Metals White Paint 0.1 0 Optical Surface Reflector 0 0.2 0.4 0.6 0.81
Non-Ideal Radiative Heat Transfer Full form of the Stefan-Boltzmann equation P = εσa T T rad ( 4 4 ) env where T env =environmental temperature (=4 K for space) Also take into account power used internally ( ) IαA + P = εσa T T 4 4 s s int rad env
Example: AERCam/SPRINT 30 cm diameter sphere α=0.2; ε=0.8 P int =200W T env =280 K (cargo bay below; Earth above) Analysis cases: Free space w/o sun Free space w/sun Earth orbit w/o sun Earth orbit w/sun
AERCam/SPRINT Analysis (Free Space) A s =0.0707 m 2 ; A rad =0.2827 m 2 Free space, no sun 4 200W Pint = εσaradt T = 8 W 0. 8 5. 67 10 0. 2827m 2 4 m K ( 2) 1 4 = 354 K
AERCam/SPRINT Analysis (Free Space) A s =0.0707 m 2 ; A rad =0.2827 m 2 Free space with sun 4 IαA + P = εσa T T = s s int rad Isα As+ P εσa rad int 1 4 = 362 K
AERCam/SPRINT Analysis (LEO Cargo Bay) T env =280 K LEO cargo bay, no sun 4 4 W Pint = Arad ( T Tenv) T = 200 εσ W 8 0. 8 5. 67 10 0. 2827m 2 4 m K LEO cargo bay with sun ( ) + 2 Isα As+ P IsαAs + P = εσa ( int rad T Tenv) T = εσa 4 4 int 4 env rad + 1 4 4 ( 280 K) = 384 K T 1 4 = 391 K
Radiative Insulation I s T insulation T wall Thin sheet (mylar/kapton with surface coatings) used to isolate panel from solar flux Panel reaches equilibrium with radiation from sheet and from itself reflected from sheet Sheet reaches equilibrium with radiation from sun and panel, and from itself reflected off panel
Multi-Layer Insulation (MLI) I s Multiple insulation layers to cut down on radiative transfer Gets computationally intensive quickly Highly effective means of insulation Biggest problem is existence of conductive leak paths (physical connections to insulated components)
1D Conduction Basic law of one-dimensional heat conduction (Fourier 1822) where K=thermal conductivity (W/m K) A=area Q = KA dt dx dt/dx=thermal gradient
3D Conduction General differential equation for heat flow in a solid r r r grt (,) ρc Trt, 2 Trt (, )+ = K K t where g(r,t)=internally generated heat ρ=density (kg/m 3 ) c=specific heat (J/kg K) K/ρc=thermal diffusivity ( )
Simple Analytical Conduction Model Heat flowing from (i-1) into (i) T i-1 T i T i+1 Q KA T T i in = x i 1 Heat flowing from (i) into (i+1) Q KA T T i+1 out = x Heat remaining in cell Q out Q = in i ρ c Ti( j+ 1) Ti( j) K t
Heat Pipe Schematic
Shuttle Thermal Control Components
Shuttle Thermal Control System Schematic
ISS Radiator Assembly