Thermal Engineering, Refrigeration & Power

In a counter-flow heat exchanger:
Temperature difference at one end (hot fluid inlet, cold fluid outlet):

$$\Delta T_1 = T_{h,\text{in}} - T_{c,\text{out}} = 150^\circ\text{C} - 50^\circ\text{C} = 100^\circ\text{C}$$

Temperature difference at the other end (hot fluid outlet, cold fluid inlet):

$$\Delta T_2 = T_{h,\text{out}} - T_{c,\text{in}} = 130^\circ\text{C} - 30^\circ\text{C} = 100^\circ\text{C}$$

When $\Delta T_1 = \Delta T_2$, the logarithmic mean temperature difference (LMTD) simplifies directly to the arithmetic difference:

$$\text{LMTD} = \Delta T_1 = \Delta T_2 = 100^\circ\text{C}$$

Option analysis:

• $50^\circ\text{C}$: Incorrect.
• $120^\circ\text{C}$: Incorrect.
• $80^\circ\text{C}$: Incorrect.
• $100^\circ\text{C}$: Correct, exact mean temperature difference.

The Grashof number ($Gr = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$) approximates the ratio of buoyant force to viscous force acting on a fluid. It is the primary dimensionless governing parameter in natural (free) convection, playing a role analogous to that of the Reynolds number in forced convection.

Option analysis:

• Prandtl number: Ratio of momentum diffusivity to thermal diffusivity ($Pr = \nu / \alpha$).
• Reynolds number: Governs forced convection and inertial/viscous flow regimes.
• Grashof number: Correct, universally characterises natural buoyant convection.
• Nusselt number: Ratio of convective to conductive heat transfer across a boundary.

In an industrial steam boiler furnace, all three modes of heat transfer take place simultaneously:

• Radiation: Direct thermal radiation from luminous flames and incandescent refractory walls to boiler water tubes.
• Convection: High-velocity hot flue gases flow over tube banks transferring heat by forced convection.
• Conduction: Heat transfers through the solid metal thickness of the boiler tube walls into the water.
  
  Option analysis:
• Steam condenser: Involves phase change convection and tube-wall conduction, negligible radiation.
• Boiler furnaces: Correct, combines intense radiation, flue gas convection, and tube-wall conduction.
• Melting of ice: Primarily conduction and natural convection.
• Electric heater: Primarily radiation and natural convection in air, not all three prominently.