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Chapter 4: Problem 53

Fluid flows through a pipe with a velocity of \(2.0 \mathrm{ft} / \mathrm{s}\) and is being heated, so the fluid temperature \(T\) at axial position \(x\) increases at a steady rate of \(30.0^{\circ} \mathrm{F} / \mathrm{min}\). In addition, the fluid temperature is increasing in the axial direction at the rate of \(2.0^{\circ} \mathrm{F} / \mathrm{ft}\) Find the value of the material derivative \(D T / D t\) at position \(x.\)

Short Answer

Expert verified
The value of the material derivative \(D T / D t\) at position \(x\) is \(4.5^{\circ} \mathrm{F} / \mathrm{s}\).

Step by step solution

01

Understand the Material Derivative

In fluid dynamics, the material derivative, denoted by \(D / Dt\), describes the rate of change of a quantity (such as temperature in this case) for a particle as it moves along with the flow. Mathematically, it is defined as the sum of the local and the convective derivatives. In this scenario, the local derivative is the rate of temperature change with respect to time, and the convective derivative is the product of velocity and the rate of temperature change with respect to axial position.
02

Convert Units

Before calculating the material derivative, make sure the units for all quantities are consistent. The rate of temperature rise per minute should be converted into seconds because the velocity is in ft/s. Therefore, convert \(30.0^{\circ} \mathrm{F} / \mathrm{min}\) to degrees Fahrenheit per second by dividing by 60, which equals \(0.5^{\circ} \mathrm{F} / \mathrm{s}\). The rate of temperature change in the axial direction and fluid velocity are both provided in terms of feet, so those can remain as is.
03

Calculate the Material Derivative

Use the definition of the material derivative to calculate \(D T / D t\). This is equal to the rate of temperature change with respect to time (\(0.5^{\circ} \mathrm{F} / \mathrm{s}\)) plus the product of the fluid velocity (\(2.0 \mathrm{ft} / \mathrm{s}\)) and the rate of temperature change with respect to axial position (\(2.0^{\circ} \mathrm{F} / \mathrm{ft}\)). This results in \(0.5 + (2.0 * 2.0) = 4.5^{\circ} \mathrm{F} / \mathrm{s}\).

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Most popular questions from this chapter

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