A long rod, insulated to prevent heat loss along its sides,...

A long rod, insulated to prevent heat loss along its sides, is in perfect thermal contact with boiling water (at atmospheric pressure) at one end and with an ice-water mixture at the other (Fig. E17.68. The rod consists of a $1.00-$ m section of copper (one end in boiling water) joined end to end to a length $L_{2}$ of steel (one end in the ice-water mixture). Both sections of the rod have cross-sectional areas of 4.00 $\mathrm{cm}^{2} .$ The temperature of the copper-steel junction is $65.0^{\circ} \mathrm{C}$ after a steady state has been set up. (a) How much heat per second flows from the boiling water to the ice-water mixture? (b) What is the length $L_{2}$ of the steel section?

Key Concepts

Composite Conduction in Multiple Materials

When a rod is made of different materials joined together, the heat transfer through the rod is determined by the combined effect of the individual thermal resistances of each material. The overall thermal resistance is treated as the sum of the individual resistances, analogous to resistors in series.

Fourier's Law of Heat Conduction

This principle states that the rate of heat transfer through a material is proportional to the negative gradient in temperature and the area through which the heat flows. It is the fundamental law used to describe how heat moves via conduction and is expressed through kA(?T/?x), where k is the thermal conductivity.

Thermal Conductivity

Thermal conductivity is an intrinsic property of materials indicating how well they can conduct heat. Different materials have different k values, which determine how efficiently they transfer heat energy when a temperature gradient is present.

Steady-State Heat Conduction

Steady-state conduction refers to a condition where the temperature distribution within the material remains constant over time. In this condition, the heat entering any segment of the material equals the heat leaving it, allowing for simplified calculations of heat flow.

Thermal Resistance

Thermal resistance is a measure of a material's opposition to heat flow. It is calculated as the ratio of the temperature difference across the material to the heat transfer rate. In composite systems, the total thermal resistance is the sum of the thermal resistances of each layer or segment.

Transcript

00:01 Okay, so this question is also on heat current.

00:04 Okay, so this question requires us to calculate how much heat per second that flows from a boiling water into an ice and water mixture through a long rod.

00:17 Now, in this situation, this long rod is basically made up of two types of metal.

00:22 One is made up of copper and then the other made up of steel.

00:26 All right.

00:26 And it's asking us to calculate the heat per second that flows from the boiling water into the ice water mixture.

00:36 So basically what is asking us is to calculate the heat current, is to calculate the heat current, is to calculate the heat current.

00:44 So let's project a diagram for our question.

00:47 So this is side with hot water and this is the side with cold water.

00:52 So we have hot water here.

00:56 And then we have water with ice.

01:02 So, okay, so this is our long road, okay, with some part of it made of copper, some part of it made up of copper, and then the other side made up of steel, which we don't know the length.

01:18 So that's it also.

01:19 Now let me do that l -s for steel.

01:22 All right.

01:23 And then the copper end is, is one meter long.

01:29 The copper end is one meter long.

01:31 So as usual, boiling water at 100 degrees celsius, then cold water at 0 degrees celsius.

01:38 Ok, now the area of this road is given us 4 .0 0 .0 cm squared.

01:46 All right, so let's move on to look at how we can solve for the heat current.

01:53 So we know that our heat current equation is given us k times area.

01:58 Into bracket, th minus t s, tc, sorry, divided by the length.

02:04 But in this situation, we don't know the length, we don't know the length of the steel.

02:11 However, let's see how best we can go about.

02:14 Now, what we need to know is that we can calculate the heat current for copper and then equated to the heat current of steel.

02:26 Because at the junction where they are connected together, they will have a thermal equilibrium.

02:32 They will all be at a thermal equilibrium.

02:34 So we can see that the heat current from copper will be able to a heat current of steel.

02:40 All right.

02:42 So picking the information for copper, we can straight away go on and calculate for the heat current of copper...

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