An ideal Carnot refrigerator (heat pump) freezes ice cubes...

An ideal Carnot refrigerator (heat pump) freezes ice cubes at the rate of $5 \mathrm{~g} / \mathrm{s}$ starting with water at the freezing point. Energy is given off to the room at $30^{\circ} \mathrm{C}$. If the fusion energy of ice is 320 joules/gram, (a) At what rate is energy expelled to the room? (b) At what rate in kilowatts must electrical energy be supplied? (c) What is the coefficient of performance of this heat pump?

An ideal Carnot refrigerator (heat pump) freezes ice cubes at the rate of $5 \mathrm{~g} / \mathrm{s}$ starting with water at the freezing point. Energy is given off to the room at $30^{\circ} \mathrm{C}$. If the fusion energy of ice is 320 joules/gram,
(a) At what rate is energy expelled to the room?
(b) At what rate in kilowatts must electrical energy be supplied?
(c) What is the coefficient of performance of this heat pump?

Key Concepts

Carnot Refrigerator

A Carnot refrigerator represents an idealized refrigeration system that operates on a reversible cycle, the Carnot cycle, to achieve the maximum possible efficiency as determined by the temperatures of the cold and hot reservoirs. This concept is fundamental in thermodynamics for understanding the limits of performance in heat removal or pumping processes.

Coefficient of Performance (COP)

The coefficient of performance (COP) for refrigeration or heat pumping is a measure of efficiency and is defined as the ratio of the heat removed from the cold reservoir (or delivered to the hot reservoir in the case of heat pumps) to the work input required to effect that transfer. It serves as a critical benchmark for comparing the performance of different refrigeration or heat pump systems.

Phase Change and Latent Heat

In processes involving phase changes—such as freezing water to form ice—the latent heat (or fusion energy) is the amount of energy required to change the phase of a substance per unit mass without changing its temperature. This concept is crucial in thermodynamics when calculating the energy involved in the phase transition, independent of temperature changes.

Heat Transfer and Energy Rates

Understanding energy transfer rates involves quantifying the amount of energy exchanged per unit time, which is typically measured in watts or kilowatts. This concept connects the principles of thermodynamics to practical applications by relating mass flow rates and energy per unit mass to overall system performance, such as the rate of heat rejected or work input in refrigeration cycles.

Transcript

00:02 Okay, a freezer has a coefficient of performance of 2 .4.

00:05 It's to convert 1 .8 kilograms of water at 25c to 1 .8 of ice at minus 1 .5 in 1 hour.

00:14 What amount of heat must be removed from the water at 25c to convert it to ice at minus 5 degrees c? okay, so, oh, and then i'll read the second part, i guess.

00:25 How much electrical energy is consumed by the freezer in this hour and see how.

00:30 How much wasted heat is delivered to the room in which the freezer sits.

00:36 Okay.

00:38 So i went ahead and wrote down the givens, the coefficient of performance k, 2 .4, and i wrote down some givens that will need, because this is a heat transfer problem, will need heat capacity.

00:51 So, and the two substances we're dealing with are water and ice, and they have different heat capacities, even though it's the same molecule.

00:58 In addition to the heat capacities, we also have this latent heat of fusion.

01:06 So it costs energy or i guess energy is released when ice is stays at zero degrees, but goes from water to ice.

01:19 Excuse me, when water stays at zero and goes from water to ice.

01:22 And then finally, i want to just note that the change in temperature of the ice.

01:28 So once, maybe it's more, it's better to read water first.

01:31 The change of temperature of the water is minus 25 degrees c because we start at 25 and we want to take it to zero.

01:38 So we need to figure out how much heat is lost in that process.

01:42 And then we'll do the phase transition, figure out how much heat is lost in that process.

01:49 But after that we need the, how much the temperature of the ice is changing.

01:54 So it's going below freezing.

01:55 So it's going minus five degrees.

01:56 Degrees c.

01:57 So that'll that'll come up and maybe the context will be more clear when i start the problem.

02:01 And i also want to note that minus 5 degrees c is minus 5k, if you're talking about temperature differences and the same for the minus 25 c and k.

02:11 Okay, great.

02:12 So now that we have all the givens written down, part a, we want to get the heat lost.

02:18 So that is just going to be the heat lost due to first get the water down.

02:26 Oh, did i write down the map? oops, i didn't.

02:28 All right, let me add that to the end...

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