Heat and Mass Transfer (ME-701) - Unit 1 Short Notes

UNIT 1: Heat and Mass Transfer – Comprehensive Short Notes

(Aligned with RGPV Past Paper Analysis)

I. Fundamentals and Basic Concepts

Modes of Heat Transfer

1. Conduction: Transfer due to temperature gradient in a medium (Fourier’s law).

2. Convection: Transfer between solid surface and moving fluid (Newton’s law of cooling).

3. Radiation: Transfer via electromagnetic waves (Stefan-Boltzmann law).

Basic Laws

Thermal Properties

• Thermal conductivity ($k$): W/m·K, material’s ability to conduct heat.

• Convective heat transfer coefficient ($h$): W/m²·K, depends on flow, geometry.

• Emissivity ($\varepsilon$): Ratio of actual to black body radiation (0–1).

• Absorptivity ($\alpha$), Reflectivity ($\rho$), Transmissivity ($\tau$): $$\displaystyle \alpha + \rho + \tau = 1 $$.

Key Distinctions

Radiation Shield

• Thin, low-emissivity material placed between surfaces to reduce net radiation exchange.

• Effectiveness: $$\displaystyle \varepsilon_{eff} \approx \frac{1}{1/\varepsilon_1 + 1/\varepsilon_2 - 1} $$ for two parallel plates.

• Application: Thermocouples, spacecraft insulation, building windows.

Thermocouple

• Principle: Seebeck effect – two dissimilar metals generate voltage proportional to temperature difference.

• Use: Temperature measurement; junction at measurement point, reference at known T.

[!TIP]

• Common Pitfall: Confusing thermal energy (stored) with heat (transfer).

• Exam Focus: Kirchhoff’s law applies at thermal equilibrium for same wavelength/direction.

II. Steady-State Heat Conduction

One-Dimensional Conduction

Plane Wall

• Governing Equation: $$\displaystyle \frac{d^2T}{dx^2} = 0 $$ (steady, 1D, constant $k$).

• Temperature Distribution: Linear: $$\displaystyle T(x) = C_1 x + C_2 $$.

• Heat Transfer Rate: $$\displaystyle Q = k A \frac{T_1 - T_2}{L} $$.

• Composite Wall:

$$Q = \frac{A (T_{hot} - T_{cold})}{\sum \frac{L_i}{k_i} + \frac{1}{h_i} + \frac{1}{h_o}} = U A \Delta T_{overall}$$

where $$\displaystyle U = \left[ \sum \frac{L_i}{k_i} + \frac{1}{h_i} + \frac{1}{h_o} \right]^{-1} $$.

• Thermal Contact Resistance: Additional resistance at interfaces due to imperfect contact.

Cylinder

• Governing Equation (radial, steady): $$\displaystyle \frac{1}{r} \frac{d}{dr} \left( r \frac{dT}{dr} \right) = 0 $$.

• Temperature Distribution: $$\displaystyle T(r) = C_1 \ln r + C_2 $$.

• Heat Transfer Rate:

$$Q = \frac{2\pi k L (T_i - T_o)}{\ln(r_o/r_i)}$$

• Critical Radius of Insulation:

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

For $$\displaystyle r < r_{cr} $$, adding insulation increases heat loss; for $$\displaystyle r > r_{cr} $$, it decreases loss.

Physical Significance: Balance between increased area and increased thermal resistance.

Sphere

• Governing Equation: $$\displaystyle \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dT}{dr} \right) = 0 $$.

• Temperature Distribution: $$\displaystyle T(r) = -\frac{C_1}{r} + C_2 $$.

• Heat Transfer Rate:

$$Q = \frac{4\pi k r_i r_o (T_i - T_o)}{r_o - r_i}$$

• Critical Radius:

$$\boxed{r_{cr} = \frac{2k}{h}}$$

Applications

• Composite Walls: Windows, cold storage, oven walls – calculate $Q$ and interface temperatures.

• Critical Radius Design:

  • Refrigeration pipes: Insulation thickness > $$\displaystyle r_{cr} $$ to minimize heat gain.

  • Electrical wires: Insulation < $$\displaystyle r_{cr} $$ to avoid overheating (but usually wires operate below $$\displaystyle r_{cr} $$).

• Plot of Q vs. r for Cylinder:

  • Initially rises (area effect dominates), peaks at $$\displaystyle r_{cr} $$, then falls (resistance effect dominates).

[!TIP]

• Critical Radius Formula: Cylinder: $k/h$; Sphere: $2k/h$.

• Common Error: Forgetting that $$\displaystyle r_{cr} $$ applies to outer radius of insulation.

III. Transient Heat Conduction

Lumped Capacitance Method

• Assumption: Infinite thermal conductivity → uniform temperature throughout solid at any time.

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

• Energy Balance:

$$\rho V c_p \frac{dT}{dt} = -h A_s (T - T_\infty)$$

• Solution:

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

where $$\displaystyle Fo = \frac{\alpha t}{L_c^2} $$ (Fourier number).

• Time to Reach Temperature $T$:

$$t = -\frac{\rho V c_p}{h A_s} \ln \frac{T - T_\infty}{T_i - T_\infty}$$

Infinite Thermal Conductivity Method

• Same as lumped capacitance; term sometimes used interchangeably.

• Temperature-Time Plot: Exponential decay from $$\displaystyle T_i $$ to $$\displaystyle T_\infty $$.

[!TIP]

• Biot Number: $$\displaystyle Bi < 0.1 $$ required; if violated, use Heisler charts or numerical methods.

• Exam Problem: Often given $\rho$, $$\displaystyle c_p $$, $k$, $h$, $V$, $$\displaystyle A_s $$ – first check $Bi$.

IV. Convection Heat Transfer

Boundary Layer Theory

• Hydrodynamic Boundary Layer: Region where velocity changes from 0 to $$\displaystyle U_\infty $$.

  • Momentum Equation Derivation (for flat plate, steady, incompressible, constant $\mu$):

    Start from Navier-Stokes, apply boundary layer approximations ($\partial/\partial x \ll \partial/\partial y$, $u \approx u(x,y)$, $v \ll u$), simplify to:

$$u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2}$$

• Thermal Boundary Layer: Region where $T$ changes from $$\displaystyle T_s $$ to $$\displaystyle T_\infty $$.

  • Energy Equation:

$$u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2}$$

• Boundary Layer Thickness ($\delta$): $$\displaystyle u/\delta \sim U_\infty/\delta $$, $$\displaystyle \delta \propto \sqrt{\frac{\nu x}{U_\infty}} $$ (laminar).

• Displacement Thickness ($$\displaystyle \delta^* $$): $$\displaystyle \delta^* = \int_0^\infty \left(1 - \frac{u}{U_\infty}\right) dy $$.

Dimensionless Numbers

Flow Regimes

• Laminar: Smooth, $$\displaystyle Re_x < 5 \times 10^5 $$ (flat plate).

• Transition: $$\displaystyle 5 \times 10^5 < Re_x < 3 \times 10^6 $$.

• Turbulent: Chaotic, $$\displaystyle Re_x > 3 \times 10^6 $$.

Diagram:

DiagramSEARCH: boundary layer development flat plate laminar turbulent

Forced Convection

External (Flow over surfaces)

• Flat Plate (Laminar):

$$Nu_x = 0.332 \, Re_x^{1/2} \, Pr^{1/3} \quad (Pr > 0.6)$$

• Cylinder/Sphere: Use Churchill-Bernstein or Hilpert correlations.

  Example (cylinder, $$\displaystyle Re_D Pr > 0.2 $$):

$$Nu_D = C \, Re_D^m \, Pr^{1/3}$$

with $C$, $m$ from table (e.g., $$\displaystyle Re_D=40–4000 $$: $$\displaystyle C=0.683 $$, $$\displaystyle m=0.466 $$).

Internal (Flow in tubes)

• Hydrodynamically Fully Developed: Velocity profile invariant in flow direction.

• Thermally Fully Developed: Temperature profile invariant.

• Entrance Length:

  • Laminar: $$\displaystyle L_{h,th} \approx 0.05 \, Re \, Pr \, D $$ (thermal).

  • Turbulent: $$\displaystyle L_{h,th} \approx 10–60 \, D $$.

• Correlation (Laminar, constant $$\displaystyle T_s $$):

$$Nu = 3.66 \quad \text{(fully developed)}$$

Dimensional Analysis for Forced Convection

Buckingham Pi Theorem: If $n$ variables involve $r$ fundamental dimensions, then $(n-r)$ independent $\pi$ terms describe the relationship.
Application to Forced Convection:

• Variables: $$\displaystyle h, L, k, \mu, c_p, \rho, U $$. ($$\displaystyle n=7 $$)

• Dimensions: $M, L, T, \Theta$ ($$\displaystyle r=4 $$).

• Repeating Variables: $L, k, \mu, \rho$ (include all dimensions).

• $\pi$ Terms:

  $$\displaystyle \pi_1 = \frac{h L}{k} = Nu $$, $$\displaystyle \pi_2 = \frac{\rho U L}{\mu} = Re $$, $$\displaystyle \pi_3 = \frac{\mu c_p}{k} = Pr $$.

• Functional Form:

$$\boxed{Nu = f(Re, Pr)}$$

Natural Convection

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

$$Nu_x = 0.59 \, (Ra_x)^{1/4} \quad \text{or} \quad Nu_x = 0.68 + \frac{0.67 \, Ra_x^{1/4}}{[1+(0.492/Pr)^{9/16}]^{4/9}}$$

• Horizontal Cylinder (average):

$$Nu_D = C \, Ra_D^m$$

(e.g., $$\displaystyle 10^4 < Ra_D < 10^7 $$: $$\displaystyle C=0.53 $$, $$\displaystyle m=1/4 $$).

[!TIP]

• Forced vs. Natural: Check $$\displaystyle Gr/Re^2 $$: if $$\displaystyle > 0.1 $$, natural convection significant.

• Correlation Selection: Ensure geometry, $Re$, $Pr$ range match.

V. Radiation Heat Transfer

Fundamental Concepts

• Black Body: Ideal emitter/absorber ($$\displaystyle \varepsilon = \alpha = 1 $$).

• Gray Body: $\varepsilon$, $\alpha$ constant (independent of $\lambda$), $$\displaystyle <1 $$.

Radiation Between Surfaces

• Shape Factor (View Factor) $$\displaystyle F_{i\to j} $$: Fraction of radiation leaving surface $i$ that strikes $j$ directly.

• Properties:

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

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

  3. Symmetry: $$\displaystyle F_{i\to j} = F_{i\to j'} $$ if $j$, $j'$ symmetric w.r.t. $i$.

Reciprocity Theorem Derivation

Consider differential areas $$\displaystyle dA_i $$, $$\displaystyle dA_j $$:

$$dF_{i\to j} = \frac{\cos \theta_i \cos \theta_j}{\pi S^2} dA_j$$

Integrate:

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

Swap integration order:

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

Thus, $$\displaystyle \boxed{A_i F_{i\to j} = A_j F_{j\to i}} $$.

Shape Factor Derivations

1. Parallel Plates (equal area $A$, spaced $L$ apart, $L \gg$ dimensions):

$$F_{1\to2} = 1, \quad F_{2\to1} = 1$$

2. Perpendicular Plates (edge-to-edge):

$$F_{1\to2} = \frac{1}{2} \left[ 1 + \frac{X^2+Y^2-1}{XY\sqrt{1+X^2+Y^2}} \right]$$

where $$\displaystyle X = a/L $$, $$\displaystyle Y = b/L $$.

3. Cylinder to Coaxial Disk:

$$F_{1\to2} = \frac{1}{2} \left[ X \sqrt{X^2-1} - \ln(X + \sqrt{X^2-1}) \right]$$

where $$\displaystyle X = L/r + 1 $$.

Net Radiation Exchange

• Between Black Bodies ($i$ and $j$):

$$Q_{i\to j} = \frac{\sigma (T_i^4 - T_j^4)}{\frac{1}{A_i F_{i\to j}}}$$

• Between Gray Bodies (using Radiosity $$\displaystyle J_i $$ and Irradiation $$\displaystyle G_i $$):

$$J_i = \varepsilon_i \sigma T_i^4 + (1-\varepsilon_i) G_i$$

$$G_i = \sum_{j=1}^N F_{i\to j} J_j$$

Solve linear system for $$\displaystyle J_i $$, then $$\displaystyle Q_i = A_i (J_i - G_i) $$.

Special Cases

• Hemispherical Cavity (radius $R$, emissivity $\varepsilon$, inner surface $T$):

  Radiation leaving opening = black body radiation from a surface of area $$\displaystyle \pi R^2 $$:

$$Q = \varepsilon \sigma (\pi R^2) T^4$$

(since cavity acts like black body for $L \gg R$).

• Large Parallel Plates (1 & 2, $$\displaystyle \varepsilon_1 $$, $$\displaystyle \varepsilon_2 $$):

$$Q_{12} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1} \cdot A$$

[!TIP]

• Radiation Shield: Two shields reduce heat transfer to $\approx 1/3$ of single-plate case.

• Common Error: Forgetting reciprocity when $$\displaystyle A_i \neq A_j $$.

VI. Heat Exchangers

Classification

LMTD Method

• Definition:

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

where $$\displaystyle \Delta T_1 $$, $$\displaystyle \Delta T_2 $$ are temperature differences at each end.

• Parallel Flow:

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

• Counter Flow:

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

• Derivation: From energy balance and $$\displaystyle \frac{dT}{dx} \propto \Delta T $$, integrate:

$$Q = U A \Delta T_{lm}$$

Design/Analysis

Given $U$, $A$, flow rates, inlet $T$:

1. Calculate $$\displaystyle C_{min} $$, $$\displaystyle C_{max} $$ ($$\displaystyle C = \dot{m} c_p $$).

2. Compute $$\displaystyle \Delta T_1 $$, $$\displaystyle \Delta T_2 $$ based on flow arrangement.

3. Find $$\displaystyle \Delta T_{lm} $$.

4. $$\displaystyle Q = U A \Delta T_{lm} = C_{min} (T_{c,out} - T_{c,in}) $$ (or hot side).

5. Solve for unknown exit temperature.

Fouling

• Definition: Deposition of scales, corrosion products on heat transfer surfaces.

• Effects: Increases thermal resistance → $U$ decreases, $Q$ drops, pressure drop increases.

• Fouling Factor $$\displaystyle R_f $$: Added resistance in $$\displaystyle 1/U = 1/U_{clean} + R_f $$.

Temperature Profiles

• Parallel Flow: Both fluids approach same exit T; $\Delta T$ decreases along length.

• Counter Flow: Can achieve outlet $$\displaystyle T_{c,out} > T_{h,out} $$; $\Delta T$ more uniform.

• Equal Slope Condition: When $$\displaystyle C_h = C_c $$ (heat capacity rates equal), temperature lines are parallel in counter flow.

[!TIP]

• LMTD Correction: For cross flow, use correction factor $F$: $$\displaystyle Q = U A F \Delta T_{lm} $$.

• Exam Tip: Always sketch temperature profiles to identify $$\displaystyle \Delta T_1 $$, $$\displaystyle \Delta T_2 $$.

VII. Extended Surfaces (Fins)

Purpose & Applications

• Increase surface area to enhance heat transfer from small base areas.

• Used in: radiators, engine cylinders, electronic cooling, heat exchangers.

Types

• Straight (rectangular, cylindrical), annular, pin.

Governing Equation (Straight Fin, Uniform Cross-Section)

Assumptions: steady, 1D, constant $k$, $h$, no radiation.

• Energy Balance on differential element:

$$-k A_c \frac{dT}{dx} + h P (T - T_\infty) dx = \frac{d}{dx} \left( -k A_c \frac{dT}{dx} dx \right)$$

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

• Simplified:

$$\frac{d^2T}{dx^2} - m^2 (T - T_\infty) = 0$$

where $$\displaystyle \boxed{m^2 = \frac{h P}{k A_c}} $$.

Boundary Conditions

1. Base: $$\displaystyle T(0) = T_b $$.

2. Tip:

   • Convective: $$\displaystyle -k A_c \left. \frac{dT}{dx} \right|_{x=L} = h A_t (T_L - T_\infty) $$.

   • Insulated: $$\displaystyle \left. \frac{dT}{dx} \right|_{x=L} = 0 $$.

   • Infinite Fin: $$\displaystyle T(L) = T_\infty $$.

Solution & Performance Parameters

• Temperature Distribution (insulated tip):

$$\frac{T(x) - T_\infty}{T_b - T_\infty} = \frac{\cosh[m(L-x)]}{\cosh(mL)}$$

• Fin Efficiency $$\displaystyle \eta_f $$:

$$\eta_f = \frac{Q_{actual}}{Q_{if\,no\,fin}} = \frac{\tanh(mL)}{mL} \quad \text{(insulated tip)}$$

$$\eta_f = \frac{\sinh(mL)}{\cosh(mL) + (h/(m k)) \sinh(mL)} \quad \text{(convective tip)}$$

• Fin Effectiveness $$\displaystyle \varepsilon_f $$:

$$\varepsilon_f = \frac{Q_{actual}}{Q_{without\,fin}} = \eta_f \frac{A_f}{A_b}$$

where $$\displaystyle A_f = A_b + P L $$ (rectangular).

[!TIP]

• Efficiency Limit: $$\displaystyle \eta_f \approx 1/(mL) $$ for $$\displaystyle mL > 2 $$; fin too long → diminishing returns.

• Common Mistake: Using $mL$ with $L$ as total length, not exposed length.

VIII. Mass Transfer

Analogy with Heat Transfer

Modes

• Diffusion: Molecular motion (Fick’s laws).

• Convection: Bulk motion + diffusion.

Fick’s Laws

• First Law (steady diffusion): $$\displaystyle j'' = -D \frac{dC}{dx} $$.

• Second Law (transient): $$\displaystyle \frac{\partial C}{\partial t} = D \nabla^2 C $$.

Convective Mass Transfer

• Mass Transfer Coefficient $$\displaystyle h_m $$:

$$j'' = h_m (C_s - C_\infty)$$

• Analogy (Chilton-Colburn):

$$St_h \cdot Pr^{2/3} = St_m \cdot Sc^{2/3}$$

where $$\displaystyle St_m = \frac{h_m}{U} $$, $$\displaystyle Sc = \nu/D $$.

Binary Mixtures

For ideal gas: $$\displaystyle p_i V = n_i R T $$ → $$\displaystyle C_i = \frac{p_i}{R T} $$ (molar concentration).

• Mass Density $$\displaystyle \rho_i = C_i M_i $$.

• Mass Fraction $$\displaystyle Y_i = \frac{\rho_i}{\rho} = \frac{p_i M_i}{p M_{avg}} $$.

• Molar Fraction $$\displaystyle x_i = \frac{p_i}{p} $$.
  Example (O₂/N₂, $$\displaystyle p=1.1 $$ bar, $$\displaystyle T=15^\circ $$C, $$\displaystyle p_{O_2}:p_{N_2}=0.21:0.79 $$):

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

2. $$\displaystyle C_{O_2} = p_{O_2}/(R T) $$, $$\displaystyle R=8314 $$ J/kmol·K, $$\displaystyle T=288 $$ K.

3. $$\displaystyle M_{O_2}=32 $$, $$\displaystyle M_{N_2}=28 $$ → $$\displaystyle M_{avg} = 0.21\times32 + 0.79\times28 = 28.84 $$ kg/kmol.

4. $$\displaystyle \rho_{O_2} = C_{O_2} M_{O_2} $$, etc.

General Equation of Mass Diffusion (Stationary Medium)

For species $A$ in mixture:

$$\frac{\partial C_A}{\partial t} = D \nabla^2 C_A + \text{source/sink terms}$$

Derived from species continuity + Fick’s first law.

[!TIP]

• Mass Average Velocity: $$\displaystyle v = \sum Y_i v_i $$; used in convective term.

• Exam Problem: Often ask to compute $$\displaystyle x_i $$, $$\displaystyle Y_i $$, $$\displaystyle \rho_i $$ from partial pressures.

IX. Dimensional Analysis

Buckingham Pi Theorem

• Statement: If a physical relation involves $n$ variables and $r$ fundamental dimensions, the relation can be expressed as a function of $(n-r)$ independent dimensionless $\pi$ terms.

• Steps:

  1. List all variables ($n$).

  2. Choose $r$ repeating variables (include all dimensions, independent).

  3. Form $\pi$ terms: $$\displaystyle \pi = (\text{non-repeating}) \times (\text{repeating})^a $$.

  4. Determine exponents by dimensional homogeneity.

  5. Write functional relation: $$\displaystyle F(\pi_1, \pi_2, ...) = 0 $$.

Application to Forced Convection (as in Section IV)

• Variables: $$\displaystyle h, L, k, \mu, c_p, \rho, U $$ → $$\displaystyle n=7 $$, $$\displaystyle r=4 $$ ($M, L, T, \Theta$) → $3$ $\pi$ terms.

• Repeating: $L, k, \mu, \rho$.

• $$\displaystyle \pi_1 = h L / k = Nu $$, $$\displaystyle \pi_2 = \rho U L / \mu = Re $$, $$\displaystyle \pi_3 = \mu c_p / k = Pr $$.

• Relation: $$\displaystyle \boxed{Nu = f(Re, Pr)} $$.

Dimensionless Groups

• $Re$: Flow regime.

• $Pr$: Relative thickness of velocity/thermal boundary layers.

• $Nu$: Convective/conductive heat transfer.

• $Gr$, $Ra$: Natural convection strength.

• $$\displaystyle Sc = \nu/D $$: Mass transfer analog of $Pr$.

• $$\displaystyle Sh = h_m L / D $$: Mass transfer analog of $Nu$.

Advantages & Limitations

[!TIP]

• Choosing Repeating Variables: Must include all dimensions; avoid dimensionless groups.

• Common Pitfall: Using $h$, $T$ as repeating variables (they are dependent).

X. Boiling and Condensation

Boiling

• Pool Boiling Curve (surface $$\displaystyle T_s $$ vs. $q''$):

  1. Natural Convection ($$\displaystyle T_s < T_{sat} $$): $q''$ increases with $\Delta T$.

  2. Nucleate Boiling: Bubbles form at nucleation sites; $q''$ high, $\Delta T$ moderate (5–30°C).

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

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

  5. Film Boiling: Stable vapor film; $q''$ increases again with $\Delta T$ (Leidenfrost effect).

• Mechanisms:

  • Nucleate: Micro-convection, latent heat removal by bubbles.

  • Film: Vapor film insulates; heat transfer by conduction/radiation through vapor.

Condensation

• Film Condensation:

  • Nusselt’s Theory (laminar film, vertical plate):

$$q'' = 0.943 \left[ \frac{\rho_l (\rho_l - \rho_v) g h_{fg} k_l^3 \Delta T}{\mu_l L} \right]^{1/4}$$

for vertical plate of height $L$.  

• Horizontal Tube:

$$q'' = 0.725 \left[ \frac{\rho_l (\rho_l - \rho_v) g h_{fg} k_l^3 \Delta T}{\mu_l D} \right]^{1/4}$$

• Dropwise Condensation:

  • Droplets form, coalesce, fall; bare surface exposed → higher $h$ (5–10× film).

  • Requires hydrophobic surface treatment; hard to maintain.

Comparison

[!TIP]

• Boiling Curve: Remember regimes and CHF.

• Condensation: Nusselt’s formulas for vertical plate/tube are key.

END OF UNIT 1 NOTES
These notes synthesize all high-frequency topics from RGPV past papers (2023–2025). Focus on boxed formulas and derivations for 8-mark questions.