African solar systems part 6: heat losses

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Introduction

Just as what goes up must come down, what gets hot must eventually cool off. As the receiver for the CSP heats up it will inevitably begin to lose heat via radiative, convective, and conductive losses. The rate for all forms of heat loss increase with the temperature of the receiver, and so there is a maximum temperature a receiver can achieve with a given amount of focused solar radiation.

The question

A spherical mirror has a radius of curvature of 2 m and an arc length of 0.5 m. The mirror is pointed towards the sun. All the incident solar radiation (at $1370~W/m^2$ ) is focused onto a perfect spherical blackbody of radius 5 cm. What is the highest temperature in Kelvin the blackbody can reach due to radiative heat loss if the ambient temperature is 298 K?

Any receiver must be attached to the ground eventually. Therefore conductive losses can also be important. The thermal efficiency of heat engines (which we may want to attach to our CSP to produce electricity or mechanical work) increases with a higher temperature heat reservoir. A typical target temperature for a CSP is $250^\circ$ C. With bad design, such temperatures may be unachievable without large mirrors as the radiative and conductive heat losses would prevent the necessary temperatures from being reached.

The answer is 630.6.

1 solution

Jatin Yadav
Apr 2, 2014

Let $R$ be radius of mirror, $\sigma$ be stefan's constant, $r$ be radius of black body, and $T$ be the max. achievable temperature.

Let $\theta$ be half of the angle subtended at center,

$R \times 2\theta = l = 0.5$

$\theta = 0.125$

Now, available area = $\pi R^2 \sin^2 \theta$

Hence, available power = $1370 \times \pi R^2 \sin^2 \theta$ .

Power lost as radiation = $\sigma \times 4 \pi r^2 (T^4 - 298^4)$

Clearly, at saturation,

$1370 \times \pi R^2 \sin^2 \theta = \sigma \times 4 \pi r^2 (T^4 - 298^4)$

Put values to obtain $T = 630.585 K$ .

Can you post a solution to the flight of a housefly if solved please

Milun Moghe - 7 years, 2 months ago

Hi, I don't know what's the problem, I am unable to write a solution as no solution box appears. It is a past brilliant problem that I had solved. Please help , staff!

jatin yadav - 7 years, 2 months ago

African solar systems part 6: heat losses

The answer is 630.6.

1 solution

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