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Question #42243

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The Question

An ideal, inviscid fluid flows steadily through a horizontal pipe of constant cross-sectional area $A$ . The fluid enters the pipe at $x=0$ with velocity $v_0$ , temperature $T_0$ , and density $ho_0$ . As the fluid flows, it is heated such that its temperature increases linearly, reaching $T_0 + \Delta T$ at $x=L$ . The coefficient of volume expansion of the fluid is $\gamma$ (assume $\gamma \Delta T \ll 1$ ). Assuming the density variation is solely due to the temperature change, which of the following expressions correctly represents the pressure drop $\Delta P = P(0) - P(L)$ across the pipe?

Options

A.
$\rho_0 v_0^2 \gamma \Delta T$
B.
$\frac{1}{2} \rho_0 v_0^2 \gamma \Delta T$
C.
$2 \rho_0 v_0^2 \gamma \Delta T$
D.
$\rho_0 v_0^2 \gamma \Delta T \left(1 + \frac{1}{2}\gamma \Delta T\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Equation of Continuity|Bernoulli’s Energy Balance|Steady State Temperature Gradient

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize the limitations of the standard Bernoulli equation ( $P + \frac{1}{2}\rho v^2 = \text{constant}$ ).

The standard form is strictly valid only for incompressible flows where density $\rho$ is constant.

When a temperature gradient induces a variable density, students often fall into the trap of applying the standard Bernoulli equation between the two ends: $P_0 + \frac{1}{2}\rho_0 v_0^2 = P_L + \frac{1}{2}\rho_L v_L^2$ .

Substituting $\rho_L \approx \rho_0(1 - \gamma \Delta T)$ and $v_L \approx v_0(1 + \gamma \Delta T)$ into this incorrect assumption yields a pressure drop of $\frac{1}{2}\rho_0 v_0^2 \gamma \Delta T$ , which is exactly half of the true value.

The correct approach requires integrating the differential form of the mechanical energy balance (Euler's equation) coupled with the mass flux conservation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Equation of Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free