Problem 39 - Physics

Question

Two objects are maintained at constant temperatures, one hot and one cold. Two identical bars can be attached end to end, as in part \(a\) of the drawing, or one on top of the other, as in part \(b\). When either of these arrangements is placed between the hot and the cold objects for the same amount of time, heat \(Q\) flows from left to right. (a) Is the area through which the heat flows greater for arrangement \(a\) or arrangement \(b ?\) (b) Is the thickness of the material through which the heat flows greater for arrangement \(a\) or arrangement \(b ?(\mathrm{c})\) Is \(Q_{a}\) less than, greater than, or equal to \(Q_{b} ?\)

Step-by-Step Solution

Answer

(a) Arrangement b. (b) Arrangement a. (c) Greater in arrangement b.

Step 1: Understanding Heat Flow Arrangements

In arrangement (a), the bars are laid end to end, which means the heat must traverse both bars sequentially. In arrangement (b), the bars are stacked, side by side, doubling the area through which heat can flow.

Step 2: Comparing Heat Flow Areas

For arrangement (a), the area through which heat flows remains as the cross-sectional area of one bar. In arrangement (b), since the bars are side by side, the effective area is doubled. Thus, the area is greater in arrangement (b).

Step 3: Comparing Heat Flow Thickness

In arrangement (a), the thickness of the material is the total length of both bars (since the heat flows sequentially through them). For arrangement (b), heat flow occurs through the thickness of only one bar. Thus, the thickness is greater in arrangement (a).

Step 4: Evaluating Heat Flow Quantity

Using the formula for heat conduction, we express heat flow as \( Q = \frac{k \, A \, (T_{hot} - T_{cold}) \, t}{d} \), where \( k \) is the thermal conductivity, \( A \) is the cross-sectional area, \( t \) is the time, and \( d \) is the thickness. Given that \( A \) is greater and \( d \) is less in arrangement (b), more heat, \( Q_b > Q_a \), flows in arrangement (b).

Key Concepts

Heat Transfer

Heat transfer occurs when thermal energy moves from one object or substance to another. This transfer happens due to a temperature difference between the two objects, where heat will always flow from a hotter area to a cooler one. In the context of our exercise, we have two objects at constant, different temperatures - one hot and the other cold. The challenge involves understanding how heat moves between these objects through two possible setups of connecting bars.

To measure and predict heat transfer precisely, we rely on a formula that considers several factors: the thermal conductivity of the material, the area through which the heat flows, the difference in temperature between the two objects, the time for which this process occurs, and the thickness of the material. Each of these factors plays an integral role in determining the rate and amount of heat transferred.

Conduction

Conduction is the process by which heat is directly transmitted through a substance when there is a temperature gradient. This is the key mechanism of heat transfer in our exercise, where the bars act as conduits for thermal energy between the hot and cold objects.

Think of conduction as a relay race, where molecules pass energy to their neighbors. The atoms and molecules in the hot end of the bar vibrate vigorously, colliding with nearby particles and transferring energy until it moves throughout the entire bar. This energy transfer continues until thermal equilibrium is reached, meaning there is no longer a temperature difference driving the heat flow.

In our setups, the bars can be aligned either end to end or stacked. The physical arrangement of the bars changes the path and efficiency of conduction.

Cross-sectional Area

The cross-sectional area involved in heat transfer is crucial because it determines how much of the bar is available for the passage of heat. In other words, it's the section of the bar that the heat actually travels through.

In arrangement (a), the bars are aligned in a line, end-to-end, which means heat passes through a single bar's width at any time. Therefore, the cross-sectional area is just that of one bar. However, in arrangement (b), where the bars are stacked, there are two widths available, effectively doubling the cross-sectional area.

• A larger cross-sectional area allows more heat to flow simultaneously, much like how a wider highway can accommodate more traffic.

Consequently, arrangement (b) provides more pathway for heat flow, explaining why this setup results in a greater heat transfer rate.

Thermal Resistance

Thermal resistance is a property of a material that describes how well it can resist the flow of heat. It is somewhat analogous to electrical resistance, but in the realm of heat transfer.

Mathematically, thermal resistance is defined as the thickness of the material divided by the product of its thermal conductivity and the area through which heat flows. This relationship implies that:

• As thickness increases, thermal resistance increases, hindering heat flow.
• As the thermal conductivity or area increases, thermal resistance decreases, facilitating heat flow.

In arrangement (a), increased thickness means higher thermal resistance, akin to having a longer barrier for the heat to traverse. This is why arrangement (b), with reduced thickness (only one bar to cross), experiences less resistance, allowing more efficient heat flow, further amplifying the heat transfer observed in that setup.