For a given value T_H (source temperature) for a reversed Carnot cycle, the variation of T_L (sink temperature) for different values of COP is represented by which one of the following graphs?

Concept:

A reversed Carnot cycle yield a refrigeration cycle, whose COP is given by:

\(COP = \frac{{Desired\;effect}}{{work\;input}}\;\;\; = \;\frac{{{T_L}}}{{{T_H} - {T_L}}}\)

where T_L = temperature of the sink and T_H = temperature of the source.

Explanation:

\(COP = \;\frac{{{{\rm{T}}_{\rm{L}}}}}{{{{\rm{T}}_{\rm{H}}} - {{\rm{T}}_{\rm{L}}}}}{\rm{\;\;}} = \frac{1}{{\frac{{{{\rm{T}}_{\rm{H}}}}}{{{{\rm{T}}_{\rm{L}}}}} - 1}}\)

As TL increases, the ratio \(\frac{T_H}{T_L}\) decreases, therefore COP increases.

Slope = \(\frac{d(COP)}{d(T_L)}\)

\(\frac{d(COP)}{d(T_L)}=\frac{(T_H\;-\;T_L)\times 1-(T_L)\times (-1)}{(T_H-T_L)^2}\)

\(\frac{d(COP)}{d(T_L)}= \;\frac{{{{\rm{T}}_{\rm{H}}}}}{{{{\rm{(T}}_{\rm{H}}} - {{\rm{T}}_{\rm{L}}}})^2}{\rm{\;\;}}\)

i.e. slope increase as TL increases.

\(COP = \frac{{Desired\;effect}}{{work\;input}}\;\;\; = \;\frac{{{T_L}}}{{{T_H} - {T_L}}}\)

Also, if we assume COP in y-axis and TL in x-axis then the above formula will look like

\(COP = \frac{{Desired\;effect}}{{work\;input}}\;\;\; = \;\frac{{{x}}}{{{y} - {x}}}\)

Thus when x = 0, y = 0 i.e. COP will be zero when TL is zero.

According to the above relation, the best-suited curve is option B, i.e. passing through origin and increasing.