Heat and Mass Transfer (ME-701) - Unit 4 Short Notes

UNIT 4: Heat and Mass Transfer – Comprehensive Short Notes

I. Introduction and Basic Principles

Core Definitions

• Heat (Q): Energy in transit due to a temperature difference. Not a property of a system.

• Internal Energy (U): Total energy (kinetic + potential) of molecules within a system. A property.

• Thermal Energy: The portion of internal energy associated with the random motion of molecules.

• Temperature (T): A measure of the average kinetic energy of molecules. The driving potential for heat transfer.

Thermodynamics vs. Heat Transfer

Modes of Heat Transfer

1. Conduction: Microscopic energy exchange via molecular vibrations/phonons & free electrons. Requires physical contact. Occurs in solids, liquids, gases.

2. Convection: Macroscopic energy transfer via bulk fluid motion (advection) + conduction within the fluid. Requires fluid motion.

3. Radiation: Energy transfer via electromagnetic waves. Does not require a medium. Occurs at speed of light.

Fundamental Laws

• Fourier's Law (Conduction): $$\displaystyle q'' = -k \frac{dT}{dx} $$ (1D). Heat flux proportional to temperature gradient.

• Newton's Law of Cooling (Convection): $$\displaystyle q = h A (T_s - T_\infty) $$. q is heat rate, h is convective coefficient.

• Stefan-Boltzmann Law (Radiation): $$\displaystyle E_b = \sigma T^4 $$ for a black body. $$\displaystyle \sigma = 5.67 \times 10^{-8} \ \mathrm{W/m^2K^4} $$.

• Kirchhoff's Law: For an opaque body in thermal equilibrium, emissivity (ε) = absorptivity (α) at a given wavelength and direction.

System of Units & Dimensional Homogeneity

• SI Units: Temperature (K), Heat (J), Heat Flux (W/m²), Heat Rate (W).

• Dimensional Homogeneity: Every term in an equation must have the same fundamental dimensions (M, L, T, Θ). A check for equation correctness.

[!TIP] Exam Alert: Distinguish clearly between heat (energy in transit) and internal energy (property). Be prepared to state all four laws with their mathematical forms.

II. Conduction Heat Transfer

A. Steady-State Conduction

Governing Equation (1D, no gen): $$\displaystyle \frac{d}{dx} \left( k \frac{dT}{dx} \right) = 0 $$ → For constant k: $$\displaystyle \frac{d^2T}{dx^2} = 0 $$. Solution is linear: $$\displaystyle T(x) = C_1 x + C_2 $$.

1. Plane Wall:

$$ q = \frac{k A (T_1 - T_2)}{L} \quad \text{or} \quad q = \frac{\Delta T}{R_{cond}} \quad \text{where} \quad R_{cond} = \frac{L}{kA} $$

Thermal Resistance Analogy: Direct analogy to Ohm's law ($$\displaystyle V=IR $$). Total resistance for series: $$\displaystyle R_{tot} = R_1 + R_2 + ... $$. For parallel: $$\displaystyle 1/R_{tot} = 1/R_1 + 1/R_2 + ... $$.

2. Composite Wall with Convection:

Overall U-value:

$$ \frac{1}{U A} = \frac{1}{h_i A} + \sum \frac{L_j}{k_j A} + \frac{1}{h_o A} $$

Problem Pattern: Given U, ΔT_total, find ΔT across a specific layer. Use $$\displaystyle q = U A \Delta T_{total} = \frac{\Delta T_{layer}}{R_{layer}} $$.

3. Cylinder & Sphere:

• Cylinder (radius r): $$\displaystyle q = \frac{2\pi k L (T_1 - T_2)}{\ln(r_2/r_1)} $$. $$\displaystyle R_{cond} = \frac{\ln(r_2/r_1)}{2\pi k L} $$.

• Sphere (radius r): $$\displaystyle q = \frac{4\pi k r_1 r_2 (T_1 - T_2)}{(r_2 - r_1)} $$. $$\displaystyle R_{cond} = \frac{(r_2 - r_1)}{4\pi k r_1 r_2} $$.

Critical Radius/Thickness of Insulation

Concept (Insulation Paradox): For a cylindrical or spherical pipe/wire with external convection, adding insulation increases the outer radius r_o. While R_cond increases, R_conv decreases because surface area A_o increases. There exists a critical r_o where total thermal resistance is minimum, and heat loss is maximum.

Derivation (Cylinder):

Total Resistance: $$\displaystyle R_{tot} = R_{conv} + R_{cond} = \frac{1}{h_o (2\pi r_o L)} + \frac{\ln(r_o/r_i)}{2\pi k_i L} $$

For minimum R_tot, set $$\displaystyle \frac{dR_{tot}}{dr_o} = 0 $$:

$$ \boxed{r_{cr,cyl} = \frac{k_i}{h_o}} \quad \text{(Critical radius)} $$

Critical Thickness: $$\displaystyle t_{cr} = r_{cr} - r_i $$.

For Sphere: $$\displaystyle \boxed{r_{cr,sph} = \frac{2k_i}{h_o}} $$.

Plot of Heat Loss vs. Outer Radius:

• Region I (r_o < r_cr): Heat loss increases with insulation (decreasing R_tot).

• Point II (r_o = r_cr): Maximum heat loss.

• Region III (r_o > r_cr): Heat loss decreases with insulation (increasing R_tot).

Applications: Electrical cables (avoid insulation to prevent overheating), refrigeration pipes (insulate to reduce loss), hot water lines.

[!TIP] Exam Alert: CRITICAL RADIUS IS FOR CYLINDERS/SPHERES ONLY. For a plane wall, adding insulation always reduces heat loss. Remember the formulas: k/h for cylinder, 2k/h for sphere.

B. Transient Conduction

1. Lumped Capacitance Method (LCM)

• Assumption: Temperature within solid is spatially uniform (Bi < 0.1).

• Biot Number Criterion: $$\displaystyle \mathrm{Bi} = \frac{h L_c}{k} < 0.1 $$. L_c = V/A_s (characteristic length).

• Solution:

$$ \frac{T(t) - T_\infty}{T_i - T_\infty} = \exp\left( -\frac{h A_s}{\rho V c_p} t \right) = \exp(-b t) $$

where $$\displaystyle b = \frac{h A_s}{\rho V c_p} $$.

• Plot: Semi-log plot of $$\displaystyle (T - T_\infty) $$ vs. t is a straight line with slope -b.

2. Infinite Thermal Conductivity Method (Heisler Charts)

• Used when Bi > 0.1. Assumes infinite k inside solid → surface temperature instantly equals fluid temperature. Not physically accurate but used as a limiting case in chart derivations.

3. Semi-Infinite Solid

• Approximation for short times before other boundaries are felt.

• Solution (constant surface temp T_s): $$\displaystyle \frac{T(x,t) - T_s}{T_i - T_s} = \mathrm{erf}\left( \frac{x}{2\sqrt{\alpha t}} \right) $$.

• Error function (erf) is tabulated.

[!TIP] Exam Alert: First step in any transient problem: Calculate Biot number. If Bi < 0.1 → use LCM. If Bi > 0.1 → mention Heisler charts/semi-infinite solution. LCM problems (e.g., steel ball quenching) are very frequent.

C. Extended Surfaces (Fins)

Purpose: Increase surface area A to enhance heat transfer from a base surface when h is small or A is limited.

Governing Differential Equation (Straight Rectangular Fin, 1D, constant k, h):

1. Energy Balance on a differential element dx at x:

   • Conduction in: $$\displaystyle -k A_c \frac{dT}{dx} |_x $$

   • Conduction out: $$\displaystyle -k A_c \frac{dT}{dx} |_{x+dx} $$

   • Convection loss: $$\displaystyle h P dx (T - T_\infty) $$

2. Net accumulation = 0 (steady state):

$$ k A_c \frac{d^2T}{dx^2} dx - h P (T - T_\infty) dx = 0 $$

3. Let $$\displaystyle \theta = T - T_\infty $$. Define $$\displaystyle m^2 = \frac{hP}{kA_c} $$.

$$ \boxed{\frac{d^2\theta}{dx^2} - m^2 \theta = 0} $$

Boundary Conditions & Solutions:

1. Base (x=0): $$\displaystyle \theta(0) = \theta_b = T_b - T_\infty $$.

2. Tip Conditions:

   • a) Insulated Tip (most common): $$\displaystyle \frac{d\theta}{dx} |_{x=L} = 0 $$

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

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

     Solution: $$\displaystyle \theta(x) = \theta_b \frac{\cosh[m(L-x)] + (h/(m k))\sinh[m(L-x)]}{\cosh(mL) + (h/(m k))\sinh(mL)} $$

   • c) Infinite Fin: $ \theta(L) \to 0 $

     Solution: $$\displaystyle \theta(x) = \theta_b e^{-mx} $$

Fin Performance Metrics:

• Fin Heat Transfer Rate: $$\displaystyle q_f = \sqrt{h P k A_c} \ \theta_b \ \frac{\sinh(mL) + (h/(mk))\cosh(mL)}{\cosh(mL) + (h/(mk))\sinh(mL)} $$ (general). For insulated tip: $$\displaystyle q_f = \sqrt{h P k A_c} \ \theta_b \ \tanh(mL) $$.

• Fin Efficiency: $$\displaystyle \eta_f = \frac{q_f}{h A_f \theta_b} $$. For insulated tip: $$\displaystyle \eta_f = \frac{\tanh(mL)}{mL} $$.

• Fin Effectiveness: $$\displaystyle \epsilon_f = \frac{q_f}{h A_b \theta_b} $$.

• Overall Fin Efficiency (for N fins): $$\displaystyle \eta_o = 1 - \frac{N A_f}{A_{b,total}} (1 - \eta_f) $$.

[!TIP] Exam Alert: Derivation of the fin equation is a MUST. Know the definitions of η_f and ε_f. For an insulated tip fin, remember the simple formulas: q_f = √(hPkA_c) θ_b tanh(mL) and η_f = tanh(mL)/(mL). Problems often ask for heat loss from a given fin geometry.

III. Convection Heat Transfer

A. Fundamentals

Boundary Layer: Region near surface where velocity (δ) and temperature (δ_t) change rapidly.

• Displacement Thickness (δ):* Distance by which outer streamline is displaced due to boundary layer.

• Momentum Thickness (θ): Related to drag force.

Key Dimensionless Numbers:

Flow Regimes:

• Laminar: Re < ~2300 (pipes), orderly flow, parabolic velocity profile.

• Transition: 2300 < Re < 4000.

• Turbulent: Re > 4000, chaotic mixing, flatter velocity profile, higher h.

Newton's Law of Cooling: $$\displaystyle q = h A (T_s - T_\infty) $$. h is not a fluid property; it depends on flow, geometry, Re, Pr.

[!TIP] Exam Alert: Be able to define all dimensionless numbers and state their physical significance. Know typical values: Air (Pr ≈ 0.7), Water (Pr ≈ 5-7), Liquid Metals (Pr << 1).

B. Forced Convection

1. External Flow (Flat Plate)

• Laminar (local): $$\displaystyle \mathrm{Nu}_x = 0.332 \ \mathrm{Re}_x^{1/2} \ \mathrm{Pr}^{1/3} $$ (for Pr > 0.6).

• Laminar (average): $$\displaystyle \overline{\mathrm{Nu}}_L = 0.664 \ \mathrm{Re}_L^{1/2} \ \mathrm{Pr}^{1/3} $$.

• Turbulent: $$\displaystyle \mathrm{Nu}_x = 0.0296 \ \mathrm{Re}_x^{4/5} \ \mathrm{Pr}^{1/3} $$.

2. External Flow (Cylinder & Sphere)

• Use average Nusselt number correlations.

• Churchill-Bernstein (valid for all Re):

$$ \overline{\mathrm{Nu}} = 0.3 + \frac{0.62 \ \mathrm{Re}^{1/2} \ \mathrm{Pr}^{1/3}}{\left[1 + (0.4/\mathrm{Pr})^{2/3}\right]^{1/4}} \left[1 + \left(\frac{\mathrm{Re}}{282000}\right)^{5/8}\right]^{4/5} $$

3. Internal Flow (Tubes)

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

• Thermal Entry Length: $$\displaystyle L_{th} \approx 0.05 \ \mathrm{Re} \ \mathrm{Pr} \ D $$ (laminar), $ \approx 10 \ D $ (turbulent).

• Fully Developed Flow: Velocity/temperature profiles no longer change in flow direction.

• Correlations (Turbulent, smooth tube, constant wall temp):

  • Dittus-Boelter (Heating): $$\displaystyle \mathrm{Nu} = 0.023 \ \mathrm{Re}^{0.8} \ \mathrm{Pr}^{n} $$, n=0.4 for heating, 0.3 for cooling.

  • Sieder-Tate (accounts for viscosity variation): $$\displaystyle \mathrm{Nu} = 0.027 \ \mathrm{Re}^{0.8} \ \mathrm{Pr}^{1/3} \left(\frac{\mu}{\mu_s}\right)^{0.14} $$.

• Laminar: Exact solutions exist. For constant wall temp: $$\displaystyle \mathrm{Nu} = 3.66 $$ (fully developed).

[!TIP] Exam Alert: For tube flow, always check if entry length effects are significant (compare L with L_entry). Use appropriate correlation based on flow regime and thermal boundary condition. Dittus-Boelter is most common for turbulent.

C. Natural Convection

• Driven by buoyancy (density differences).

• Governing parameter: Rayleigh number (Ra = Gr·Pr).

• Vertical Plate (Laminar): $$\displaystyle \overline{\mathrm{Nu}}_L = 0.59 \ (\mathrm{Ra}_L)^{1/4} $$ for $$\displaystyle 10^4 < \mathrm{Ra}_L < 10^9 $$.

• Vertical Plate (Turbulent): $$\displaystyle \overline{\mathrm{Nu}}_L = 0.10 \ (\mathrm{Ra}_L)^{1/3} $$ for $$\displaystyle 10^9 < \mathrm{Ra}_L < 10^{13} $$.

• Horizontal Cylinder: $$\displaystyle \overline{\mathrm{Nu}}_D = C \ (\mathrm{Ra}_D)^n $$ (C and n depend on Ra range).

[!TIP] Exam Alert: Natural convection problems: Calculate Gr and Pr → get Ra → select correct correlation based on geometry and Ra range → find h → find q.

D. Phase Change Convection

1. Boiling

• Boiling Curve (q'' vs. ΔT = T_s - T_sat):

  1. Natural Convection: Single-phase, no bubbles.

  2. Nucleate Boiling: Bubbles form at nucleation sites. High heat transfer coefficient.

  3. Critical Heat Flux (CHF): Maximum q'' before film boiling. Caused by vapor blanket blocking liquid.

  4. Transition Boiling: Unstable, oscillatory.

  5. Film Boiling: Stable vapor film. Leidenfrost point. Low heat transfer coefficient.

• Mechanism: In nucleate boiling, bubbles agitate liquid, enhancing h.

2. Condensation

• Film Condensation: Vapor condenses into liquid film that flows down. Nusselt's Analysis for vertical plate: $$\displaystyle q'' = 0.943 \ [k^3 \rho_l (\rho_l - \rho_v) g h_{fg}]^{1/4} \ (\Delta T)^{3/4} $$.

• Dropwise Condensation: Vapor condenses into discrete droplets that roll off, exposing fresh surface. h_filmwise << h_dropwise (up to 10x higher). Difficult to maintain (requires hydrophobic surface, noncondensibles absent).

[!TIP] Exam Alert: Know the boiling curve regimes and why CHF occurs. Film vs. dropwise condensation comparison is a very frequent short note question. Dropwise is more effective but hard to sustain.

IV. Radiation Heat Transfer

A. Fundamentals

• Thermal Radiation: EM waves emitted by matter due to temperature (λ ≈ 0.1-100 μm).

• Black Body: Ideal surface that absorbs all incident radiation (α=1) and is the best emitter at a given T. Emissive power: $$\displaystyle E_b = \sigma T^4 $$.

• Gray Body: Surface with constant, wavelength-independent emissivity ε < 1. Real surfaces are often approximated as gray.

• Radiation Properties (Opaque Surface, τ=0):

$$ \alpha + \rho = 1 \quad \text{and} \quad \varepsilon = \alpha \ \text{(Kirchhoff, at equilibrium)} $$

B. Radiation Laws

• Planck's Law: Spectral emissive power of a black body: $$\displaystyle E_{b\lambda} = \frac{C_1}{\lambda^5 (e^{C_2/(\lambda T)} - 1)} $$.

• Wien's Displacement Law: $$\displaystyle \lambda_{max} T = b $$ (constant ≈ 2898 μm·K). Peak wavelength shifts with T.

• Stefan-Boltzmann Law: Total emissive power: $$\displaystyle E_b = \sigma T^4 $$.

• Lambert's Cosine Law: Radiation intensity is proportional to cosine of angle from normal: $$\displaystyle I = I_n \cos\theta $$.

C. Radiation Exchange Between Surfaces

1. Shape Factor (View Factor) $$\displaystyle F_{i\to j} $$

• Definition: Fraction of radiation leaving surface i that strikes surface j directly.

• Properties:

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

  • Summation: $$\displaystyle \sum_{j=1}^{N} F_{i\to j} = 1 $$.

  • Symmetry: For symmetric geometries, $$\displaystyle F_{i\to j} = F_{i\to k} $$ if j and k are symmetric.

2. Common Shape Factor Derivations (Must Know):

• Parallel, Equal, Opposing Plates (separation D): $$\displaystyle F_{1\to2} = 1 $$.

• Concentric Cylinders (long): $$\displaystyle F_{1\to2} = 1 $$, $$\displaystyle F_{2\to1} = \frac{A_1}{A_2} = \frac{r_1}{r_2} $$.

• Concentric Spheres: $$\displaystyle F_{1\to2} = 1 $$, $$\displaystyle F_{2\to1} = \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 $$.

• Cylinder to its Base: $$\displaystyle F_{\text{cyl}\to\text{base}} = \frac{1}{2} \left[ 1 + \frac{L}{\sqrt{L^2 + (D/2)^2}} \right]^{-1} $$ (for finite cylinder).

• Surface to Itself: $$\displaystyle F_{i\to i} = 0 $$ for convex surfaces; >0 for concave.

3. Radiosity (J) and Irradiation (G)

• Radiosity (J): Total radiation leaving a surface per unit area (emitted + reflected). $$\displaystyle J = \varepsilon E_b + \rho G $$.

• Irradiation (G): Total radiation incident on a surface per unit area from all others.

• For gray, diffuse surface: $$\displaystyle J = \varepsilon \sigma T^4 + (1-\varepsilon) G $$.

4. Net Radiation Exchange (Two Surfaces, Gray, Diffuse)

• General Form (using resistances):

$$ Q_{1\to2} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1-\varepsilon_1}{A_1 \varepsilon_1} + \frac{1}{A_1 F_{1\to2}} + \frac{1-\varepsilon_2}{A_2 \varepsilon_2}} $$

• For Black Bodies (ε=1): $$\displaystyle Q_{12} = \sigma A_1 F_{1\to2} (T_1^4 - T_2^4) $$.

5. Radiation Shields

• Thin, highly reflective (low ε) surface placed between two radiating bodies.

• Effect: For N shields, net heat transfer reduces by factor ≈ 1/(N+1).

• Mechanism: Each shield intercepts radiation and re-radiates, creating multiple resistances in series.

[!TIP] Exam Alert: Shape factor problems are very frequent. Use reciprocity and summation to find unknown F. The net exchange formula with resistances is key. For two black bodies, use the simple form. Radiation shield effect is a common conceptual question.

V. Heat Exchangers

A. Classification

• By Flow Pattern:

  • Parallel Flow (Co-current): Both fluids flow same direction. ΔT decreases along exchanger.

  • Counter Flow: Fluids flow opposite directions. Can achieve ΔT_min → more efficient. ΔT can be constant.

  • Cross-Flow: Fluids flow perpendicular. Often used with fins.

• By Construction:

  • Double-Pipe: Simplest, one pipe inside another. Used for small capacities.

  • Shell-and-Tube: One shell, many tubes. Versatile, handles high P/T. (Most common industrial).

  • Plate: Gasketed plates. Compact, good for low-viscosity fluids.

  • Air-Cooled: Fins on tubes, air blown across. Used where water scarce.

B. Analysis Methods

1. Log Mean Temperature Difference (LMTD)

• Derivation (Parallel Flow): From energy balance, q = C_c (T_{c,out} - T_{c,in}) = C_h (T_{h,in} - T_{h,out}). Temperature profiles are linear. q = U A ΔT_lm.

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

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

• For Counter Flow: Same formula, but ΔT_1 = T_{h,in} - T_{c,out}, ΔT_2 = T_{h,out} - T_{c,in}.

• Correction Factor (F): For cross-flow or multi-pass shell-and-tube, use ΔT_lm,F = F * ΔT_lm. F is a function of P and R (capacity ratio and temperature ratio).

2. Effectiveness-NTU Method (Brief)

• Effectiveness: $$\displaystyle \epsilon = \frac{q}{q_{max}} = \frac{\text{Actual heat transfer}}{\text{Max possible heat transfer}} $$.

• Max possible: $$\displaystyle q_{max} = C_{min} (T_{h,in} - T_{c,in}) $$.

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

• Relations: For counter flow: $$\displaystyle \epsilon = \frac{1 - \exp[-\mathrm{NTU}(1-C_r)]}{1 - C_r \exp[-\mathrm{NTU}(1-C_r)]} $$, where $$\displaystyle C_r = C_{min}/C_{max} $$.

3. Condition for Equal Slopes (Counter Flow)

• When $$\displaystyle C_{min} = C_{max} $$ (i.e., $$\displaystyle C_r = 1 $$), temperature profiles are parallel (equal slopes). ΔT is constant along exchanger. Effectiveness: $$\displaystyle \epsilon = \frac{\mathrm{NTU}}{1 + \mathrm{NTU}} $$.

[!TIP] Exam Alert: LMTD calculation is extremely frequent. Always draw temperature diagram to identify ΔT_1 and ΔT_2 correctly. For cross-flow, remember to find F from charts. Heat exchanger problems typically give inlet/outlet temps, flow rates, U, find A or unknown temp.

VI. Mass Transfer

A. Fundamentals

1. Fick's Laws

• First Law (Steady, 1D): $$\displaystyle J_A = -D \frac{dC_A}{dx} $$. J_A = diffusion flux (mol/m²s), D = diffusivity.

• Second Law (Transient): $$\displaystyle \frac{\partial C_A}{\partial t} = D \frac{\partial^2 C_A}{\partial x^2} $$. Analogous to transient heat conduction.

2. General Equation of Mass Diffusion (Stationary Medium)

• Derivation (Control Volume):

  • Accumulation = Net diffusive inflow.

  • For 1D: $$\displaystyle \frac{\partial C}{\partial t} \Delta x = J_x|_x A - J_x|_{x+\Delta x} A $$

  • $$\displaystyle \frac{\partial C}{\partial t} = -\frac{\partial J_x}{\partial x} $$

  • Substitute Fick's First Law: $$\displaystyle \frac{\partial C}{\partial t} = \frac{\partial}{\partial x} \left( D \frac{\partial C}{\partial x} \right) $$

  • For constant D: $$\displaystyle \boxed{\frac{\partial C}{\partial t} = D \nabla^2 C} $$.

3. Mixture Properties Calculations (Binary Mixture)

Given: Partial pressures $$\displaystyle P_A, P_B $$, Total P, Temperature T.

• Molar Concentration (C): $$\displaystyle C = \frac{P}{RT} $$ (total or partial: $$\displaystyle C_A = P_A / RT $$).

• Mass Density: $$\displaystyle \rho_i = \frac{P_i M_i}{RT} $$, Total $$\displaystyle \rho = \sum \rho_i $$.

• Mass Fraction: $$\displaystyle w_i = \frac{\dot{m}_i}{\dot{m}_{total}} = \frac{\rho_i}{\rho} $$.

• Molar Fraction: $$\displaystyle y_i = \frac{P_i}{P} = \frac{C_i}{C} $$.

4. Mass Average Velocity

• When mass transfer occurs, bulk fluid moves. Mass average velocity v is the velocity at which the total mass flows. For binary mixture: $$\displaystyle \mathbf{v} = \frac{\dot{m}_A \mathbf{v}_A + \dot{m}_B \mathbf{v}_B}{\dot{m}_A + \dot{m}_B} $$.

• Significance: Diffusion flux relative to mass average velocity: $$\displaystyle J_A = \rho_A (\mathbf{v}_A - \mathbf{v}) $$.

B. Convective Mass Transfer

• Mass Transfer Coefficient (h_m): $$\displaystyle N_A = h_m (C_{A,s} - C_{A,\infty}) $$ (analogous to Newton's law).

• Chilton-Colburn Analogy: $$\displaystyle \frac{h_m}{\rho D} = \frac{h}{\rho c_p} \mathrm{Pr}^{2/3} $$ or $$\displaystyle \mathrm{St}_m \mathrm{Sc}^{2/3} = \mathrm{St} \mathrm{Pr}^{2/3} $$.

• Evaporation vs. Boiling:

  • Evaporation: Surface phenomenon. Liquid at surface vaporizes into gas. Can occur at any T.

  • Boiling: Bulk phenomenon. Vapor bubbles form within liquid at nucleation sites. Occurs at T_sat.

C. Applications

Humidification, Drying, Distillation, Absorption, Transient diffusion (e.g., doping in semiconductors).

[!TIP] Exam Alert: Mixture properties calculations (mass fraction, molar fraction, densities) are repeated problems. Practice converting between P_i, y_i, w_i, ρ_i. The diffusion equation derivation is a common 7-mark question.

VII. Dimensional Analysis and Similarity

A. Buckingham Pi Theorem

• Statement: If a physical relation involves n variables and k fundamental dimensions (M, L, T, Θ), then the relation can be expressed as a function of n-k independent dimensionless π-groups: $$\displaystyle F(\pi_1, \pi_2, ..., \pi_{n-k}) = 0 $$.

• Procedure:

  1. List all n variables and their dimensions.

  2. Choose k repeating variables that together include all fundamental dimensions (usually from the list: ρ, μ, c_p, k, D, V, L).

  3. Form n-k π-groups by combining each remaining variable with the repeating variables. Make each π-group dimensionless.

  4. Express functional relationship: $$\displaystyle \pi_1 = f(\pi_2, \pi_3, ...) $$.

Example (Forced Convection, Nu = f(Re, Pr)):

Variables: h, k, ρ, μ, c_p, V, L → n=7, k=3 (M,L,Θ) → n-k=4 π-groups.

Choose repeating: ρ, V, L (include M,L; need Θ? Add c_p or k). Standard choice: ρ, V, L, μ (but μ has M,L,T). Better: ρ, V, L, c_p? Let's derive:

• π₁: h with ρ, V, L, c_p → h / (ρ c_p V) = St.

• π₂: μ with ρ, V, L → μ/(ρ V L) = 1/Re.

• π₃: k with ρ, c_p, V, L → k/(ρ c_p V L) = 1/(Re Pr).

• π₄: D (if mass transfer)...

But we know Nu = hL/k, Re = ρVL/μ, Pr = μ c_p / k. So Nu = f(Re, Pr) is the final correlation.

B. Important Dimensionless Groups (Recap)

• Heat Transfer: Re, Pr, Nu, Gr, Ra, St, Pe.

• Mass Transfer: Sc (Schmidt) = ν/D (analog of Pr), Sh (Sherwood) = h_m L / D (analog of Nu).

C. Application to Heat Sink (May 2024 Problem)

Variables: k, h, t, L, ΔT, ρ, c_p, μ. n=8, k=3 (M,L,Θ) → 5 π-groups.

Choose repeating: ρ, μ, L (cover M,L,T? μ has M,L,T. ρ has M,L³. L has L. Need Θ? Use ΔT or c_p). Standard for forced convection fins: h, k, t, L, ΔT, ρ, c_p, μ.

Possible π-groups:

1. π₁ = hL/k = Nu (heat transfer).

2. π₂ = ρ V L / μ = Re (flow). But V not in list? Problem says "variables involved are...". V might be implied. If not, we have t (thickness). Need to be careful.

   • If V is not a variable, we must use given ones. Repeating: ρ, μ, L. Then:

   • π₁ (with h): h L / k → Nu.

   • π₂ (with t): t/L → Geometric ratio.

   • π₃ (with ΔT): c_p ΔT / (something)? This gets tricky. The key is to form groups that make physical sense. The expected answer likely includes Nu, Re, Pr, and geometric ratios like t/L.

   • Standard Result for fins: η_f or q_f depends on mL = L √(hP/(kA_c)). So mL is a key group, which is L √(h/(k t)) for straight fin. This combines Nu (via hL/k) and geometry L/t.

[!TIP] Exam Alert: Buckingham Pi is a 8-mark derivation question. Be systematic: list variables, dimensions, choose repeating, form π-groups. Advantages: reduces experiments, groups parameters. Limitations: doesn't give functional form, may miss important physics if key variable omitted.

END OF UNIT 4 NOTES
Focus on bolded high-frequency topics from past papers: Critical radius, Fin efficiency/equation, LMTD, Shape factors/reciprocity, Mass mixture properties, Buckingham Pi derivation.