Heat and Mass Transfer (ME-701) - Unit 3 Short Notes

UNIT 3: Advanced Heat and Mass Transfer Topics

1.0 Extended Surfaces (Fins)

1.1 Introduction and Applications of Fins

Extended surfaces (fins) are protrusions attached to a base surface to increase the effective surface area for heat transfer, thereby enhancing the rate of heat dissipation or absorption.

• Applications: Heat exchangers, electronic cooling (CPU heatsinks), radiators, airfoil leading edges, cryogenic systems.

• Primary Purpose: To reduce the temperature gradient at the base by increasing the area for convection.

1.2 Fin Performance Parameters

[!TIP] Exam Tip: Fin efficiency is always ≤ 1. Fin effectiveness can be > 1. For a fin to be useful, $$\displaystyle \varepsilon_f > 1 $$.

1.3 Governing Differential Equation for One-Dimensional Fins

Assumptions: Steady-state, 1D conduction in fin, constant $k$, $h$, $$\displaystyle T_\infty $$, negligible radiation, uniform cross-section.

• General Energy Balance on a differential element:

$$q_x - q_{x+dx} - h P (T - T_\infty) dx = 0$$

where $P$ is perimeter, $$\displaystyle A_c $$ is cross-sectional area.

• Fourier's Law: $$\displaystyle q_x = -k A_c \frac{dT}{dx} $$

• Governing ODE:

$$\frac{d^2 \theta}{dx^2} - m^2 \theta = 0 \quad \text{where} \quad \theta = T - T_\infty, \quad m^2 = \frac{h P}{k A_c}$$

Boundary Conditions (for fin of length $L$):

1. At base ($$\displaystyle x=0 $$): $$\displaystyle T = T_b $$ or $$\displaystyle \theta = \theta_b $$

2. At tip ($$\displaystyle x=L $$):

   • Insulated tip: $$\displaystyle \frac{d\theta}{dx} = 0 $$

   • Convective tip: $$\displaystyle -k A_c \frac{d\theta}{dx}\big|_{x=L} = h A_f \theta(L) $$

   • Prescribed temperature: $$\displaystyle T = T_L $$

   • Infinite fin: $\theta \to 0$ as $x \to \infty$

1.4 Temperature Distribution and Heat Transfer Rate

Solution for uniform cross-section fin with insulated tip:

• Temperature: $$\displaystyle \frac{\theta}{\theta_b} = \frac{\cosh[m(L-x)]}{\cosh(mL)} $$

• Heat transfer rate: $$\displaystyle q_{fin} = \sqrt{h P k A_c} \theta_b \tanh(mL) $$

For other tip conditions, solutions involve $\sinh$, $\cosh$, or exponential terms.

Analysis of Common Fin Types:

• Rectangular (Straight): $$\displaystyle A_c = t \cdot w $$, $$\displaystyle P = 2(w + t) \approx 2w $$ (if $t \ll w$)

• Pin Fin: $$\displaystyle A_c = \frac{\pi d^2}{4} $$, $$\displaystyle P = \pi d $$, $$\displaystyle m^2 = \frac{4h}{kd} $$

• Annular Fin: $$\displaystyle A_c = \frac{\pi (r_o^2 - r_i^2)}{2} $$ (for radial coordinate $r$), $$\displaystyle P = 2\pi r_m $$ (mean radius)

[!TIP] Common Pitfall: For annular fins, the governing equation is in cylindrical coordinates: $$\displaystyle \frac{d}{dr}\left(r \frac{d\theta}{dr}\right) - \frac{h r}{k} \theta = 0 $$.

2.0 Critical Insulation Thickness

2.1 Concept and Physical Significance

For a cylindrical system (e.g., pipe, wire), adding insulation initially decreases total heat loss because conduction resistance increases. Beyond a certain critical radius, adding more insulation increases heat loss because the increased outer surface area for convection outweighs the added conduction resistance.

• Significance: In designing insulated cylinders (electrical cables, refrigeration pipes), one must ensure insulation thickness is greater than the critical radius to achieve desired heat reduction.

2.2 Derivation of Critical Radius for Cylinder

Total thermal resistance per unit length:

$$R_{tot} = R_{cond} + R_{conv} = \frac{\ln(r_o/r_i)}{2\pi k} + \frac{1}{2\pi r_o h}$$

Heat transfer rate: $$\displaystyle q = \frac{T_i - T_\infty}{R_{tot}} $$

For minimum heat loss (maximum $$\displaystyle R_{tot} $$), set $$\displaystyle \frac{dR_{tot}}{dr_o} = 0 $$:

$$\frac{d}{dr_o}\left(\frac{\ln(r_o/r_i)}{2\pi k} + \frac{1}{2\pi r_o h}\right) = \frac{1}{2\pi k r_o} - \frac{1}{2\pi h r_o^2} = 0$$

Solving:

$$\boxed{r_{cr} = \frac{k}{h}}$$

Critical insulation thickness: $$\displaystyle t_{cr} = r_{cr} - r_i = \frac{k}{h} - r_i $$

2.3 Heat Transfer vs. Outer Radius Plot

• For $$\displaystyle r_o < r_{cr} $$: $q$ decreases with increasing $$\displaystyle r_o $$ (insulation helps).

• At $$\displaystyle r_o = r_{cr} $$: $q$ is minimum.

• For $$\displaystyle r_o > r_{cr} $$: $q$ increases with $$\displaystyle r_o $$ (insulation backfires).

• Reason: Beyond $$\displaystyle r_{cr} $$, the $$\displaystyle 1/r_o $$ dependence of convective resistance dominates the $$\displaystyle \ln(r_o) $$ dependence of conductive resistance.

2.4 Applications in Engineering Design

• Electrical cables: Insulation thickness must be $$\displaystyle > k/h $$ to prevent overheating.

• Refrigeration pipes: Ensure insulation is above critical radius to minimize heat gain.

• Steam pipes: Often operate with $$\displaystyle r_o > r_{cr} $$, so adding insulation always reduces loss.

[!TIP] Exam Focus: For a given cylinder, if $$\displaystyle r_i > k/h $$, then $$\displaystyle r_{cr} < r_i $$ and no critical radius exists—adding insulation always reduces heat loss.

3.0 Transient Heat Conduction

3.1 Lumped Capacitance Method

Assumption: Temperature is spatially uniform ($$\displaystyle \theta = \theta(t) $$ only). Valid when internal conduction resistance $\ll$ external convection resistance.

• Applicability Criterion: Biot Number $$\displaystyle Bi = \frac{h L_c}{k} < 0.1 $$, where $$\displaystyle L_c = V/A_s $$.

• Energy Balance:

$$\rho V c_p \frac{d\theta}{dt} = -h A_s \theta$$

• Solution:

$$\frac{\theta}{\theta_i} = \exp\left(-\frac{h A_s}{\rho V c_p} t\right) = \exp(-Fo \cdot Bi)$$

where Fourier Number $$\displaystyle Fo = \frac{\alpha t}{L_c^2} $$, $$\displaystyle \alpha = k/(\rho c_p) $$.

• Plot: $$\displaystyle \ln(\theta/\theta_i) $$ vs. $t$ is linear with slope $$\displaystyle -hA_s/(\rho V c_p) $$.

3.2 Infinite Thermal Conductivity Method

Concept: Neglects temperature gradients within the solid (similar to lumped capacitance but used in specific contexts like semi-infinite solids with sudden surface temperature change).

• Governing Equation (for semi-infinite solid):

$$\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$$

• Boundary Conditions:

  • $$\displaystyle T(x,0) = T_i $$

  • $$\displaystyle T(0,t) = T_s $$ (constant surface temperature)

  • $$\displaystyle T(\infty,t) = T_i $$

• Solution:

$$\frac{T(x,t) - T_s}{T_i - T_s} = \operatorname{erf}\left(\frac{x}{2\sqrt{\alpha t}}\right)$$

• Temperature-Time Variation at Surface ($$\displaystyle x=0 $$): $$\displaystyle T_s $$ is constant, so no variation—this method is for prescribed surface temperature, not convection.

[!TIP] Distinguish: Lumped capacitance assumes uniform $T$ throughout; infinite conductivity is a misnomer for semi-infinite solid with prescribed $$\displaystyle T_s $$. In some curricula, "infinite conductivity" refers to the limit $k \to \infty$ in transient conduction, leading to uniform temperature.

3.3 Heisler Charts

• Purpose: Graphical solution for 1D transient conduction in plane walls, cylinders, spheres.

• Variables: $$\displaystyle \theta/\theta_i $$ vs. $Fo$ for different $Bi$.

• Not explicitly asked but useful for understanding transient response when $$\displaystyle Bi > 0.1 $$.

4.0 Convection Heat Transfer

4.1 Flow Regimes and Boundary Layer Theory

• Laminar Flow: Smooth, orderly fluid motion ($$\displaystyle Re_x < 5 \times 10^5 $$ for flat plate).

• Turbulent Flow: Chaotic, mixing motion ($$\displaystyle Re_x > 5 \times 10^5 $$).

• Boundary Layer: Region near surface where velocity (hydrodynamic) and temperature (thermal) gradients are significant.

  • Displacement thickness $$\displaystyle \delta^* $$: Measure of mass flow defect.

  • Momentum thickness $\theta$: Measure of momentum defect.

  • Thermal boundary layer thickness $$\displaystyle \delta_t $$: Region where $T$ changes from $$\displaystyle T_s $$ to $$\displaystyle T_\infty $$.

  • For $Pr \approx 1$, $$\displaystyle \delta_t \approx \delta $$; for $$\displaystyle Pr > 1 $$, $$\displaystyle \delta_t < \delta $$; for $$\displaystyle Pr < 1 $$, $$\displaystyle \delta_t > \delta $$.

Momentum Equation for Flat Plate (Boundary Layer Approximations):

• Continuity: $$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$

• $x$-momentum: $$\displaystyle u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} $$

• Assumptions: Steady, incompressible, negligible pressure gradient, $$\displaystyle \frac{\partial}{\partial x} \ll \frac{\partial}{\partial y} $$.

4.2 Dimensionless Groups in Convection

4.3 Forced Convection

External Flow (Flat Plate, Cylinder):

• Local $$\displaystyle Nu_x $$ for laminar flat plate ($$\displaystyle Pr > 0.6 $$):

$$Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}$$

• Average $$\displaystyle Nu_L $$ for laminar flat plate:

$$\overline{Nu}_L = 0.664 Re_L^{1/2} Pr^{1/3}$$

• Flow over Cylinder (valid for $$\displaystyle Re_D \cdot Pr > 0.2 $$):

$$\overline{Nu}_D = C Re_D^m Pr^{0.37} \quad \text{( Churchill-Bernstein)}$$

where $C$ and $m$ depend on $$\displaystyle Re_D $$ range.

Internal Flow (Tubes):

• Hydrodynamic Entry Length: $$\displaystyle L_{hyd} \approx 0.05 Re \cdot D $$ (laminar), $10 D$ (turbulent).

• Thermal Entry Length: $$\displaystyle L_{th} \approx 0.05 Re \cdot Pr \cdot D $$ (laminar).

• Fully Developed Turbulent Flow (Dittus-Boelter):

$$Nu = 0.023 Re^{0.8} Pr^n \quad (n=0.4 \text{ for heating}, 0.3 \text{ for cooling})$$

Valid for $$\displaystyle Re > 10000 $$, $$\displaystyle 0.7 < Pr < 160 $$, $$\displaystyle L/D > 60 $$.

4.4 Natural Convection

• Driving Force: Buoyancy ($Gr$ or $Ra$).

• Vertical Plate (laminar, $$\displaystyle Ra_x < 10^9 $$):

$$Nu_x = 0.59 (Gr_x \cdot Pr)^{1/4} \quad \text{or} \quad \overline{Nu}_L = 0.68 + \frac{0.67 (Ra_L)^{1/4}}{[1 + (0.492/Pr)^{9/16}]^{4/9}}$$

• Horizontal Cylinder:

$$\overline{Nu}_D = C Ra_D^m \quad \text{(e.g., for } 10^4 < Ra_D < 10^{12}, C=0.53, m=1/4)$$

• Enclosures (vertical): $Nu$ depends on $Ra$ and aspect ratio.

4.5 Boiling and Condensation

Pool Boiling Curve (surface vs. saturation temperature):

1. Natural Convection: $q''$ increases with $\Delta T$.

2. Nucleate Boiling: Bubbles form at nucleation sites; $q''$ high.

3. Critical Heat Flux (CHF): Maximum $q''$ before film boiling.

4. Transition Boiling: Unstable, $q''$ drops.

5. Film Boiling: Stable vapor film; $q''$ lower, increases with $\Delta T$.

Condensation:

• Filmwise Condensation: Continuous film; $h$ lower.

• Dropwise Condensation: Droplets roll off; $h$ much higher (5-10×).

• Nusselt's Theory for Vertical Plate (Film Condensation):

$$q'' = 0.943 \left[ \frac{k^3 \rho_l (\rho_l - \rho_v) g h_{fg}}{\mu_l (T_{sat} - T_s)} \right]^{1/4} (T_{sat} - T_s)^{3/4}$$

[!TIP] Distinguish: Boiling occurs when liquid vaporizes from heated surface; evaporation occurs at liquid-vapor interface below saturation.

5.0 Heat Exchangers

5.1 Classification

5.2 Log Mean Temperature Difference (LMTD)

Definition: Equivalent constant $$\displaystyle \Delta T_m $$ that gives same $q$ as actual varying $\Delta T$.

• Parallel Flow:

$$\Delta T_1 = T_{h,in} - T_{c,in}, \quad \Delta T_2 = T_{h,out} - T_{c,out}$$

$$\Delta T_m = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}$$

• Counter Flow:

$$\Delta T_1 = T_{h,in} - T_{c,out}, \quad \Delta T_2 = T_{h,out} - T_{c,in}$$

Same formula for $$\displaystyle \Delta T_m $$.

• LMTD Correction Factor ($F$) for cross flow/shell-and-tube:

$$q = U A F \Delta T_m$$

$F$ depends on $$\displaystyle P = (T_{h,in} - T_{h,out})/(T_{c,out} - T_{c,in}) $$ and $$\displaystyle R = C_{min}/C_{max} $$.

5.3 Effectiveness-NTU Method

• Effectiveness: $$\displaystyle \varepsilon = \frac{q}{q_{max}} = \frac{q}{C_{min}(T_{h,in} - T_{c,in})} $$

• NTU: $$\displaystyle \text{NTU} = \frac{U A}{C_{min}} $$

• Counter Flow:

$$\varepsilon = \frac{1 - \exp[-NTU(1 - C_r)]}{1 - C_r \exp[-NTU(1 - C_r)]}, \quad C_r = C_{min}/C_{max}$$

• Parallel Flow:

$$\varepsilon = \frac{1 - \exp[-NTU(1 + C_r)]}{1 + C_r}$$

• Cross Flow (both fluids unmixed): Use chart or approximation.

5.4 Design and Performance Calculations

Steps:

1. Determine $$\displaystyle C_h $$, $$\displaystyle C_c $$, $$\displaystyle C_{min} $$, $$\displaystyle C_{max} $$, $$\displaystyle C_r $$.

2. Choose method (LMTD if both inlet/outlet temps known; ε-NTU if area known).

3. For LMTD: Compute $$\displaystyle \Delta T_1 $$, $$\displaystyle \Delta T_2 $$, $$\displaystyle \Delta T_m $$, apply $F$ if needed.

4. For ε-NTU: Compute $$\displaystyle q = \varepsilon C_{min} \Delta T_{max} $$, then $$\displaystyle T_{out} $$ from energy balance.

5. Fouling: Add fouling resistance $$\displaystyle R_f $$: $$\displaystyle \frac{1}{U_{clean}} = \frac{1}{U_{dirty}} + R_f $$.

5.5 Special Cases

• Equal Slope Condition: In counter flow, when $$\displaystyle C_h = C_c $$ ($$\displaystyle C_r = 1 $$), temperature profiles are parallel (equal slope). Then:

$$\varepsilon = \frac{NTU}{1 + NTU} \quad \text{(from limit of counter flow formula)}$$

• LMTD with Given Exit Temps: Direct substitution into $$\displaystyle \Delta T_1 $$, $$\displaystyle \Delta T_2 $$.

6.0 Radiation Heat Transfer

6.1 Fundamental Concepts

6.2 View Factors (Shape Factors)

• Definition: $$\displaystyle F_{i \to j} $$ = fraction of radiation leaving surface $i$ that strikes surface $j$ directly.

• Properties:

  1. Reciprocity: $$\displaystyle A_i F_{i \to j} = A_j F_{j \to i} $$

  2. Summation: $$\displaystyle \sum_{j=1}^N F_{i \to j} = 1 $$ (including self-view $$\displaystyle F_{i \to i} $$)

  3. Symmetry: For symmetric geometries, $$\displaystyle F_{i \to j} = F_{j \to i} $$ if $$\displaystyle A_i = A_j $$.

• General Expression for Same-Area Surfaces ($$\displaystyle A_i = A_j $$):

$$F_{i \to j} = \frac{1}{A_i} \int_{A_i} \int_{A_j} \frac{\cos \theta_i \cos \theta_j}{\pi S^2} dA_j dA_i$$

where $S$ is distance between differential areas.

• Common View Factors:

  • Parallel plates (equal area $A$, spaced $L$): $$\displaystyle F_{1 \to 2} = \frac{1}{2} \left[ X - (X^2 - 1)^{1/2} \right] $$, $$\displaystyle X = 1 + \frac{L^2}{A} $$.

  • Cylinder to coaxial cylinder: $$\displaystyle F_{1 \to 2} = 1 $$ (enclosure).

  • Cylinder to its base: $$\displaystyle F_{\text{base} \to \text{curved}} = 1 $$; $$\displaystyle F_{\text{curved} \to \text{base}} = \frac{A_{\text{base}}}{A_{\text{curved}}} $$.

[!TIP] Exam Trick: For a convex surface, $$\displaystyle F_{i \to i} = 0 $$ (cannot see itself). For concave, $$\displaystyle F_{i \to i} > 0 $$.

6.3 Radiative Exchange Between Surfaces

• Radiosity ($$\displaystyle J_i $$): Total radiation leaving surface $i$ (emitted + reflected).

$$J_i = \varepsilon_i E_{b,i} + (1 - \varepsilon_i) G_i$$

where $$\displaystyle G_i $$ = irradiation on $i$.

• Net Heat Transfer from surface $i$:

$$q_i = A_i (J_i - G_i)$$

• Network Method (for gray, diffuse surfaces):

  • Resistance Network: $$\displaystyle q_{1 \to 2} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1 - \varepsilon_1}{A_1 \varepsilon_1} + \frac{1}{A_1 F_{1 \to 2}} + \frac{1 - \varepsilon_2}{A_2 \varepsilon_2}} $$

  • For multiple surfaces, solve using radiosity equations.

6.4 Radiation Shields

• Purpose: Reduce radiative heat transfer between surfaces by inserting a low-emissivity shield.

• Effectiveness: For two shields between two surfaces, heat transfer reduces by factor $$\displaystyle \approx \frac{1}{N+1} $$ where $N$ = number of shields, if all surfaces have equal area and $\varepsilon$.

• Calculation: Extend resistance network with shield surfaces.

6.5 Radiation Laws

• Planck's Law (Spectral emissive power of black body):

$$E_{b,\lambda} = \frac{2\pi h c^2}{\lambda^5 \left( e^{hc/(\lambda k T)} - 1 \right)}$$

• Wien's Displacement Law: $$\displaystyle \lambda_{max} T = 2898 \ \mu m \cdot K $$

• Stefan-Boltzmann Law (Total emissive power):

$$E_b = \sigma T^4, \quad \sigma = 5.67 \times 10^{-8} \ \mathrm{W/m^2 K^4}$$

For gray surface: $$\displaystyle E = \varepsilon \sigma T^4 $$.

7.0 Mass Transfer

7.1 Introduction and Analogies with Heat Transfer

• Modes:

  • Molecular Diffusion: Random motion (Fick's law).

  • Convective Mass Transfer: Bulk motion + diffusion (analogous to forced convection).

  • Radiation: Negligible for mass.

• Applications: Drying, evaporation, distillation, drug delivery, air pollution.

• Analogy: Replace $T \to C$, $k \to D$, $$\displaystyle h \to h_m $$, $Pr \to Sc$, $Re$ same.

7.2 Fick's Law of Diffusion

One-Dimensional Steady-State:

$$N_A = -D \frac{dC_A}{dx} \quad \text{(molar flux)}$$

or

$$n_A = -\rho D \frac{d\omega_A}{dx} \quad \text{(mass flux)}$$

where $D$ = mass diffusivity ($$\displaystyle \mathrm{m^2/s} $$).

7.3 General Diffusion Equation in Stationary Medium

From conservation of mass for species $A$ in binary mixture ($A$ and $B$):

$$\frac{\partial C_A}{\partial t} = D \nabla^2 C_A \quad \text{(if no bulk flow, constant } D, \text{ isothermal)}$$

• Cartesian: $$\displaystyle \frac{\partial C_A}{\partial t} = D \left( \frac{\partial^2 C_A}{\partial x^2} + \frac{\partial^2 C_A}{\partial y^2} + \frac{\partial^2 C_A}{\partial z^2} \right) $$

• Cylindrical (radial only): $$\displaystyle \frac{\partial C_A}{\partial t} = D \left( \frac{\partial^2 C_A}{\partial r^2} + \frac{1}{r} \frac{\partial C_A}{\partial r} \right) $$

• Spherical (radial only): $$\displaystyle \frac{\partial C_A}{\partial t} = D \left( \frac{\partial^2 C_A}{\partial r^2} + \frac{2}{r} \frac{\partial C_A}{\partial r} \right) $$

7.4 Convective Mass Transfer

• Mass Transfer Coefficient ($$\displaystyle h_m $$): $$\displaystyle N_A = h_m (C_{A,\infty} - C_{A,s}) $$

• Chilton-Colburn Analogy:

$$j_D = j_H = \frac{St \cdot Re^{2/3}}{Pr^{2/3}} = \frac{St \cdot Re^{2/3}}{Sc^{2/3}}$$

where $$\displaystyle St = \frac{h_m}{V} $$ (mass Stanton), $$\displaystyle St_h = \frac{h}{\rho c_p V} $$ (heat Stanton).

• Correlations: Use heat transfer correlations by replacing $Nu \to Sh$, $Pr \to Sc$.

  Example: For flow over flat plate: $$\displaystyle Sh_x = 0.332 Re_x^{1/2} Sc^{1/3} $$.

7.5 Binary Mixtures

For a gas mixture of $A$ and $B$:

• Molar concentration: $$\displaystyle C_A = \frac{p_A}{RT} $$ (ideal gas)

• Mass concentration: $$\displaystyle \rho_A = \frac{M_A p_A}{RT} $$

• Molar fraction: $$\displaystyle x_A = \frac{p_A}{p_{total}} $$ (Dalton's law: $$\displaystyle p_{total} = p_A + p_B $$)

• Mass fraction: $$\displaystyle \omega_A = \frac{\rho_A}{\rho} = \frac{M_A x_A}{M_A x_A + M_B x_B} $$

• Example (O₂/N₂ at 1.1 bar, 15°C):

  $$\displaystyle p_{O_2} = 0.21 \times 1.1 = 0.231 \ \mathrm{bar} $$, $$\displaystyle p_{N_2} = 0.79 \times 1.1 = 0.869 \ \mathrm{bar} $$.

  $$\displaystyle C_{O_2} = p_{O_2}/(RT) $$, etc.

8.0 Dimensional Analysis

8.1 Buckingham Pi Theorem

Statement: If a physical problem involves $n$ variables and $r$ fundamental dimensions, then the variables can be grouped into $(n - r)$ independent dimensionless $\Pi$ groups. Procedure:

1. List all variables ($n$) and their dimensions.

2. Identify repeating variables ($r$) that include all fundamental dimensions.

3. Form $\Pi$ groups: $$\displaystyle \Pi = \frac{\text{Non-repeating variables}}{\text{Product of repeating variables raised to powers}} $$.

4. Solve for exponents by equating dimensions.

5. Express functional relation: $$\displaystyle F(\Pi_1, \Pi_2, ...) = 0 $$.

8.2 Application to Forced Convection

Variables: $$\displaystyle q = f(h, L, k, \Delta T, \rho, \mu, c_p, V) $$

Dimensions: $$\displaystyle [q] = \Theta $$, $$\displaystyle [h] = \Theta^{-1} $$, $$\displaystyle [L] = L $$, $$\displaystyle [k] = \Theta^{-1} $$, $$\displaystyle [\Delta T] = \Theta $$, $$\displaystyle [\rho] = ML^{-3} $$, $$\displaystyle [\mu] = ML^{-1}T^{-1} $$, $$\displaystyle [c_p] = L^2T^{-2}\Theta^{-1} $$, $$\displaystyle [V] = LT^{-1} $$.

Fundamental dimensions: $M, L, T, \Theta$ → $$\displaystyle r=4 $$. $$\displaystyle n=8 $$ → $$\displaystyle n-r=4 $$ $\Pi$ groups.

Choose repeating variables: $L, k, \rho, V$ (common in convection).

• $$\displaystyle \Pi_1 = \frac{q}{L^2 k \Delta T} $$ → $$\displaystyle \Pi_1 = Nu $$ (Nusselt)

• $$\displaystyle \Pi_2 = \frac{\rho V L}{\mu} $$ → $$\displaystyle \Pi_2 = Re $$ (Reynolds)

• $$\displaystyle \Pi_3 = \frac{\mu c_p}{k} $$ → $$\displaystyle \Pi_3 = Pr $$ (Prandtl)

• $$\displaystyle \Pi_4 = \frac{q}{\rho V L^2 c_p \Delta T} $$ → $$\displaystyle \Pi_4 = St $$ (Stanton)

Relation: $$\displaystyle Nu = f(Re, Pr, St) $$ but $$\displaystyle St = Nu/(Re \cdot Pr) $$, so $$\displaystyle Nu = f(Re, Pr) $$.

8.3 Advantages and Limitations

Advantages:

• Reduces number of experiments.

• Provides fundamental dimensionless groups.

• Applicable to geometrically similar systems. Limitations:

• Does not give exact functional form.

• Cannot distinguish between variables with same dimensions (e.g., $L$ and $D$).

• Choice of repeating variables affects $\Pi$ groups.

8.4 Application to Heat Sink (May 2024)

Variables: $$\displaystyle q = f(k, h, t, L, \Delta T, \rho, c_p, \mu) $$ (for air properties). $$\displaystyle n=8 $$, dimensions: $M, L, T, \Theta$ → $$\displaystyle r=4 $$.

Repeating variables: $$\displaystyle L, k, \rho, c_p $$ (or $L, k, \mu, \rho$).

Typical $\Pi$ groups:

• $$\displaystyle \Pi_1 = \frac{q}{k L \Delta T} $$ → related to $Nu$

• $$\displaystyle \Pi_2 = \frac{h L}{k} $$ → $Nu$

• $$\displaystyle \Pi_3 = \frac{\rho V L}{\mu} $$ → $Re$ (but $V$ not given? May need to include velocity or derive from geometry)

• $$\displaystyle \Pi_4 = \frac{\mu c_p}{k} $$ → $Pr$

• $$\displaystyle \Pi_5 = \frac{t}{L} $$ (geometric ratio)

• $$\displaystyle \Pi_6 = \frac{\rho V^2}{?} $$ etc. Final correlation: $$\displaystyle Nu = f(Re, Pr, t/L) $$.

9.0 Special Topics and Definitions (Recurring Short Answers)

9.1 Basic Definitions

• Newton's Law of Cooling: $$\displaystyle q = h A (T_s - T_\infty) $$.

• Heat: Energy in transit due to temperature difference.

• Internal Energy: $u$ (J/kg); total energy of molecules.

• Thermal Energy: Often synonymous with internal energy; sometimes refers to sensible/latent heat.

• Conduction vs. Convection: Conduction = molecular vibration; convection = conduction + bulk fluid motion.

9.2 Distinctions

9.3 Composite Wall and Multilayer Systems

• Overall $U$:

$$\frac{1}{U A} = \sum \frac{L_i}{k_i A_i} + \frac{1}{h_i A_i} + \frac{1}{h_o A_o}$$

For plane walls, $A$ constant: $$\displaystyle \frac{1}{U} = \sum \frac{L_i}{k_i} + \frac{1}{h_i} + \frac{1}{h_o} $$.

• Temperature Drop Across Layer $i$:

$$q = \frac{T_{i-1} - T_i}{L_i/(k_i A)} \quad \Rightarrow \quad \Delta T_i = q \cdot \frac{L_i}{k_i A}$$

• Example (Cold Storage): Compute $U$, then $$\displaystyle q = U A \Delta T_{overall} $$, then individual $$\displaystyle \Delta T_i $$.

9.4 Thermocouple

• Principle: Seebeck effect: two dissimilar metals joined at junction generate EMF proportional to temperature difference.

• Use: Temperature measurement; reference junction needed for absolute $T$.

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