ऊष्मागतिकी, ऊर्जा एवं यांत्रिकी

Detailed Analysis of Heat Exchanger Performance:

• Assertion (A) is True: A counter-flow heat exchanger delivers a much higher rate of heat transfer and thermal effectiveness than a parallel-flow exchanger for the same surface area, because the Log Mean Temperature Difference (LMTD) is significantly larger and fluid exit temperatures can cross.
• Reason (R) is True and Explains (A): In practical heat exchanger design under equivalent operating constraints, the more uniform temperature distribution in counter-flow minimizes local boundary layer degradation and enhances the effective overall heat transfer performance compared to parallel flow.

By Fourier's law of conduction, heat transfer rate is $Q = \frac{k A \Delta T}{L}$. Wall surface area $A = 5\text{ m} \times 4\text{ m} = 20\text{ m}^2$. Temperature difference $\Delta T = 110^\circ\text{C} - 40^\circ\text{C} = 70\text{ K}$. Thickness $L = 0.25\text{ m}$. Thermal conductivity $k = 0.70\text{ W/m}\cdot\text{K}$. Thus, $Q = \frac{0.70 \times 20 \times 70}{0.25} = \frac{980}{0.25} = 3920\text{ W} = 3.92\text{ kW}$. Option analysis: (A) $2.19\text{ kW}$ is incorrect. (B) $3.92\text{ kW}$ is the exact calculated heat loss rate. (C) $4.52\text{ kW}$ is incorrect. (D) $2.88\text{ kW}$ is incorrect.

Statement (I) is false: the critical radius of insulation ($r_c = k/h$) is the radius at which the total thermal resistance is minimum and heat transfer rate is MAXIMUM, not minimum. Statement (II) is true: adding insulation up to $r_c$ increases the external surface area for convection faster than conductive resistance, thereby increasing heat transfer rate. Option analysis: (A) Statement (I) is false. (B) Statement (II) is true. (C) Statement (I) is false. (D) Statement (I) is false but Statement (II) is true.

Correct definitions of fluid types in fluid mechanics:
(A) Ideal fluid: A theoretical fluid that is incompressible and has zero viscosity (\(\mu = 0\))\lt /strong\gt \(\rightarrow\) (I).
(B) Newtonian fluid: Obeys Newton's law of viscosity, where shear stress is linearly proportional to shear strain rate (\(\tau = \mu \frac{du}{dy}\))\lt /strong\gt \(\rightarrow\) (II) (e.g., water, air, light oils).
(C) Real fluid: Any actual physical fluid that possesses viscosity (\(\mu > 0\))\lt /strong\gt and exhibits friction \(\rightarrow\) (IV).
(D) Non-Newtonian fluid: Fluid where shear stress is not linearly proportional to shear rate \(\rightarrow\) \lt strong\gt (III)\lt /strong\gt (e.g., paint, starch suspensions, blood). \lt strong\gt (E) Ideal plastic fluid (Bingham plastic)\lt /strong\gt : Does not deform until a definite \lt strong\gt yield stress (\(\tau_0\)) is exceeded, after which shear stress is proportional to shear strain rate \(\rightarrow\) (V) (e.g., toothpaste, sewage sludge).