Cold water (cp=4.18 kJ / kg ·^∘ C) leading to a shower enters a well-insulated, thin-walled, dou

step 1 So here we have, let's see here, cold water. And we're given the heat capacity of the cold water

Cold water (cp=4.18 kJ / kg ·^∘ C) leading to a shower...

Cold water $\left(c_{p}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)$ leading to a shower enters a well-insulated, thin-walled, double-pipe, counterflow heat exchanger at $15^{\circ} \mathrm{C}$ at a rate of $0.25 \mathrm{kg} / \mathrm{s}$ and is heated to $45^{\circ} \mathrm{C}$ by hot water $\left(c_{p}=4.19 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)$ that enters at $100^{\circ} \mathrm{C}$ at a rate of $3 \mathrm{kg} / \mathrm{s}$. Determine $(a)$ the rate of heat transfer and $(b)$ the rate of exergy destruction in the heat exchanger. Take $T_{0}=25^{\circ} \mathrm{C}$

Cold water $\left(c_{p}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)$ leading to a shower
enters a well-insulated, thin-walled, double-pipe, counterflow heat exchanger at $15^{\circ} \mathrm{C}$ at a rate of $0.25 \mathrm{kg} / \mathrm{s}$ and is heated to $45^{\circ} \mathrm{C}$ by hot water $\left(c_{p}=4.19 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\right)$ that enters
at $100^{\circ} \mathrm{C}$ at a rate of $3 \mathrm{kg} / \mathrm{s}$. Determine $(a)$ the rate of heat transfer and $(b)$ the rate of exergy destruction in the heat exchanger. Take $T_{0}=25^{\circ} \mathrm{C}$

Key Concepts

Heat Transfer in Heat Exchangers

This concept involves the calculation of energy exchange between two fluids at different temperatures in a device designed to optimize heat exchange. It requires understanding how the rate of heat transfer depends on the fluids’ mass flow rates, specific heat capacities, and the temperature difference between the inlet and outlet streams. The analysis typically employs conservation of energy principles and heat exchanger effectiveness in determining how much heat energy is transferred per unit time.

Specific Heat Capacity

Specific heat capacity is a material property that quantifies how much energy is needed to raise the temperature of a unit mass of a substance by one degree Celsius. It is a critical parameter in calculating heat transfer because it directly relates the heat energy exchanged to the temperature change experienced by a fluid, which is essential in designing and analyzing thermal systems like heat exchangers.

Exergy Analysis

Exergy analysis focuses on assessing the quality or usefulness of energy within a system by taking into account both the quantity and the potential to perform work. Unlike energy, exergy is destroyed due to irreversibilities in processes, and its analysis is crucial for determining the efficiency of thermal systems. It helps identify where and how much energy is lost in forms that cannot do work, thereby guiding improvements in system design.

Exergy Destruction

Exergy destruction quantifies the loss of work potential due to irreversibilities such as friction, heat transfer across finite temperature differences, and mixing. In the context of thermal systems like heat exchangers, calculating exergy destruction is important for understanding how far the actual performance deviates from the ideal, reversible process as dictated by the Second Law of Thermodynamics.

Second Law of Thermodynamics

The Second Law of Thermodynamics introduces the concept of irreversibility in energy conversion processes, asserting that some energy quality is always lost as heat that cannot be converted into useful work. This law underpins the necessity for exergy analysis in thermal systems, as it provides the theoretical framework for understanding and quantifying exergy destruction caused by non-idealities in heat transfer.

Transcript

00:01 So here we have, let's see here, cold water.

00:04 And we're given the heat capacity of the cold water.

00:12 Leading to a shower, so enters a well -insid thin wall, double -piped counterflow heat exchanger.

00:21 And it enters that 15 degrees c, and a flow rate of 0 .25 kilograms per second.

00:27 So we have cold water coming in here.

00:29 Then we have hot water flowing, you know, usually.

00:32 In the opposite direction, but it doesn't really matter that much.

00:37 Flowing through here.

00:39 And then so basically we're just warming up this cold water from the hot water that's flowing through.

00:48 Not sure why you do this and not just mix them.

00:52 It's probably a good reason for doing it, though.

00:56 The hot water is at 45 degrees c, and it has a heat capacity that's a touch slower.

01:02 Now, we know, again, this is a steady state process.

01:07 So if we do this control volume, the energy flow in must equal the energy flow out.

01:14 And let's see here.

01:15 We know that, no, i think i said this wrong, that this is the temperature of the water coming out.

01:23 So we increase the temperature by 30 degrees c in this heat exchanger.

01:28 And the water coming in is, let's see here, the hot water.

01:36 Water is basically at, you know, 100 degrees c.

01:39 So it's just, you know, not boil, you know, it's just ready to boil, basically.

01:46 But it is still liquid.

01:50 And it's flowing through at a rate of three kilograms per second.

01:55 And so we want to frame the rate of heat transfer and the rate of extra destruction in the heat exchanger.

02:01 Okay.

02:02 So the energy equation tells us that the, you know, the generally says that the rate of heat flux, heat flow in, heat transfer in, plus the rate of entropy coming in equals the rate of entropy going out.

02:23 And so if we look at just the, if we do a control volume of just the cold water, just look at the cold water, we have the heat flow into the cold water, is the mass flow rate of the cold water.

02:40 It's got a one there, times the heat capacity of the cold water, times the difference in temperature from the exit to the inlet...

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