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Consider the net radiant heat flux between an object and its surroundings as given by
qrad,net = ǫ σ ( T4 − T ∞4 ) ( 1 ) where ǫ is the emissivity, σ is the Stefan-Boltzmann constant (5.670374419×10−8 in
W ), T is the object’s temperature and T is the surrounding (ambient) temperature. m2K4 ∞
1-a) Linearize qrad,net in terms of ǫ, T, and T∞, use deviation variables and call this expression qL′ .
1-b) Compute the value for qrad,net with ǫ = 0.9, T = 700 K, and T ∞ = 300 K.
1-c) Using both the linearized and nonlinear equation, compute the numeric change
in qrad,net (in watts) for each of the following cases:
(a) ǫ increases by 0.01.
(b) T increases by 1 K.
(c) T∞ increases by 1 K.
(d) T increases by 5 K.
(e) T∞ increases by 5 K.
(f) ǫ increases by 0.01 and T increases by 5 K and T∞ increases by 5 K.
(g) Using the linearized model, find the range of T that causes a ± 5% change in qrad,net.
(h) Using the nonlinear model, find the range of T that causes a ± 5% change in qrad,net.
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