For an ideal radiator (hohlraum) with a 10-cm-diameter opening, located in black surroundings at 16°C, (a) calculate the net radiant heat transfer rate for hohlraum temperatures of 100°C and 560°C, (b) the wavelength at which the emission is a maximum, (c) the monochromatic emission at ?max, and (d) the wavelengths at which the monochromatic emission is 1 per cent of the maximum value.

GIVEN

• An ideal radiator (hohlraum) in black surroundings

• Radiator opening diameter (D) = 10 cm = 0.1 m

• Surrounding temperature (Ts) = 16°C = 289 K

• Hohlraum temperatures

- Th1 = 100°C = 373 K

- Th2 = 560°C = 833 K

FIND

(a) The net radiant heat transfer rate (qr)

(b) The wavelength at which the emission is maximum (?max)

(c) The monochromatic emission at ? max (E?max)

(d) The wavelengths at which the monochromatic emission is 1% E?max

SKETCH



PROPERTIES AND CONSTANTS

the Stephan-Boltzmann constant (?) = 5.67 × 10–8 W/(m2 K4)


All parts of the problem will first be solved for Th = 373 K

(a) The net radiative transfer between any two black surfaces is







(b) The wavelength at which the maximum emission occurs for a black body is





(c) The monochromatic emission is



(d)



(There will be one solution below ?max and one above ?max)

By trial and error



Repeating the above proceduur for Th = 833 K

Physics & Space Science

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A long copper cylinder 0.6-m in diameter and initially at a uniform temperature of 38°C is placed in a water bath at 93°C. Assuming that the heat transfer coefficient between the copper and the water is 1248 W/(m2 K), calculate the time required to heat the center of the cylinder to 66°C. As a first approximation, neglect the temperature gradient within the cylinder, then repeat your calculation without this simplifying assumption and compare your results.

GIVEN
• A long copper cylinder is placed in a water bath
• Diameter of cylinder (D) = 0.6 m
• Initial temperature (To) = 38°C
• Water bath temperature (T?) = 93°C
• The heat transfer coefficient ( ) ch = 1248 W/(m2 K)
FIND
Calculate the time required to heat the center of the cylinder to 66°C assuming
(a) Negligible temperature gradient within the cylinder
(b) Without this simplification, then
(c) Compare your results
ASSUMPTIONS
• Neglect end effects
• Radial conduction only
SKETCH

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The length of a certain wire is halved while the radius is doubled. What is the change in the resistance of this wire?

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