9.2 Heat Transfer Applications

9.2 Heat Transfer Applications

1. Free and Forced Convection

1. Free (Natural) Convection:

   • Fluid motion driven by buoyancy forces due to density differences.

   • Density variation caused by temperature gradients.

   • Governing Equation: $q = hA(T_s - T_\infty)$

   • Key Parameters:

     • Grashof Number ($Gr$): Ratio of buoyancy to viscous forces. $Gr = \frac{g\beta(T_s - T_\infty)L^3}{\nu^2}$

     • Rayleigh Number ($Ra$): Product of Grashof and Prandtl numbers. $Ra = Gr \cdot Pr$

   • Flow Regimes:

     • Laminar: Smooth, orderly fluid motion.

     • Turbulent: Chaotic, swirling motion (higher heat transfer).

   • Examples: Heating of room by radiator, cooling of electronic chips.

2. Forced Convection:

   • Fluid motion induced by external means (pump, fan, wind).

   • Governing Equation: $q = hA(T_s - T_\infty)$ (with h typically higher than free convection)

   • Key Parameters:

     • Reynolds Number ($Re$): Ratio of inertial to viscous forces. $Re = \frac{\rho VL}{\mu} = \frac{VL}{\nu}$

     • Nusselt Number ($Nu$): Dimensionless heat transfer coefficient. $Nu = \frac{hL}{k}$

   • Correlations: $Nu = f(Re, Pr)$ for different geometries and flow conditions.

   • Examples: Car radiator cooling, heat exchangers, air conditioning.

3. Comparison:

   • Free Convection: Lower h-values, simpler systems, no moving parts.

   • Forced Convection: Higher h-values, requires power input, more control.

2. Fins

1. Purpose: Increase surface area for enhanced heat transfer.

2. Types:

   • Straight fins (constant cross-section)

   • Annular fins (circular, around tubes)

   • Pin fins (cylindrical rods)

3. Fin Effectiveness ($\epsilon_f$):

   • Ratio of fin heat transfer to base heat transfer without fin.

   • $\epsilon_f = \frac{q_f}{hA_b(T_b - T_\infty)}$

   • Fin is effective if $\epsilon_f > 1$.

4. Fin Efficiency ($\eta_f$):

   • Ratio of actual fin heat transfer to ideal heat transfer (if entire fin at base temperature).

   • $\eta_f = \frac{q_f}{q_{max}} = \frac{q_f}{hA_f(T_b - T_\infty)}$

   • Always less than 1.

5. Key Parameters:

   • Fin parameter: $m = \sqrt{\frac{hP}{kA_c}}$

   • Where: P = perimeter, $A_c$ = cross-sectional area, k = fin material conductivity.

6. Applications:

   • Heat sinks for electronics

   • Automotive radiators

   • Air-cooled engines

   • Condensers and evaporators

3. Heat Exchangers and Effectiveness

1. Heat Exchanger Types:

   • Parallel Flow: Fluids flow in same direction.

   • Counter Flow: Fluids flow in opposite directions (most efficient).

   • Cross Flow: Fluids flow perpendicular to each other.

   • Shell and Tube: One fluid flows inside tubes, other flows outside in shell.

   • Compact: High surface area to volume ratio.

2. Log Mean Temperature Difference (LMTD) Method:

   • For heat exchanger design: $q = UA\Delta T_{lm}$

   • $\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$

   • Valid for known inlet/outlet temperatures.

3. Effectiveness-NTU Method:

   • Used when outlet temperatures are unknown.

   • Effectiveness ($\epsilon$): Ratio of actual heat transfer to maximum possible. $\epsilon = \frac{q_{actual}}{q_{max}} = \frac{C_h(T_{h,i} - T_{h,o})}{C_{min}(T_{h,i} - T_{c,i})} = \frac{C_c(T_{c,o} - T_{c,i})}{C_{min}(T_{h,i} - T_{c,i})}$

   • Number of Transfer Units (NTU): $NTU = \frac{UA}{C_{min}}$

   • Capacity Ratio (C): $C = \frac{C_{min}}{C_{max}}$

   • Relationships: $\epsilon = f(NTU, C, flow arrangement)$

4. Key Formulas:

   • Heat transfer rate: $q = \dot{m}c_p\Delta T$ for each fluid.

   • Energy balance: $q = \dot{m}_h c_{p,h}(T_{h,i} - T_{h,o}) = \dot{m}_c c_{p,c}(T_{c,o} - T_{c,i})$

   • For counter-flow with $C = 1$: $\epsilon = \frac{NTU}{1 + NTU}$

5. Applications:

   • Power plant condensers

   • Refrigeration systems

   • Automotive radiators

   • HVAC systems

   • Chemical processing