Heat and Mass Transfer (ME-701) - Unit 5 Short Notes

UNIT 5: Heat and Mass Transfer

Focus: Exam-Centric Short Notes Based on RGPV Past Papers

1. Steady-State Heat Conduction

One-Dimensional Conduction

• Plane Wall:

$$ q = \frac{k A (T_1 - T_2)}{L}, \quad T(x) = T_1 - \frac{q}{kA} x $$

• Cylinder (radial):

$$ q = \frac{2\pi k L (T_i - T_o)}{\ln(r_o/r_i)}, \quad T(r) = T_i - \frac{q}{2\pi k L} \ln\left(\frac{r}{r_i}\right) $$

• Sphere:

$$ q = \frac{4\pi k r_i r_o (T_i - T_o)}{r_o - r_i}, \quad T(r) = T_i + \frac{q}{4\pi k} \left( \frac{1}{r} - \frac{1}{r_i} \right) $$

Composite Systems

• Series Resistances:

$$ q = \frac{\Delta T}{\sum R}, \quad R_{\text{cond}} = \frac{L}{kA} \text{ (plane)}, \quad R_{\text{cond}} = \frac{\ln(r_o/r_i)}{2\pi k L} \text{ (cylinder)} $$

• Parallel Resistances:

$$ q = q_1 + q_2, \quad \Delta T \text{ same for each path} $$

• Overall Heat Transfer Coefficient (U-value):

$$ q = U A \Delta T_{\text{overall}}, \quad \frac{1}{U} = \sum \frac{1}{h} + \sum R $$

For plane wall with convection:

$$ \frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o} $$

[!TIP] For cylindrical coordinates, use log mean area for conduction resistance: \( A_{\text{lm}} = \frac{A_o - A_i}{\ln(A_o/A_i)} \).

Multi-Layer Walls and Pipes

• Example: Double-pane window (May 2023) – total resistance includes convection films and conduction layers (glass, air gap).

• Contact Resistance: Conceptual – imperfect contact adds extra thermal resistance; significant in electronic packaging.

2. Extended Surfaces (Fins)

Purpose and Applications

• Increase surface area to enhance heat transfer from a base.

• Used in heat sinks, radiators, engine cooling, HVAC.

Performance Parameters

• Fin Efficiency:

$$ \eta_f = \frac{q_f}{h A_f (T_b - T_\infty)} $$

Measures effectiveness relative to ideal fin at base temperature.

• Fin Effectiveness:

$$ \epsilon_f = \frac{q_f}{h A_b (T_b - T_\infty)} $$

Ratio of actual heat transfer to that from base area without fin.

Governing Differential Equation (Rectangular Fin, Uniform Cross-Section)

Energy balance on differential element:

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

Using Fourier’s law \( q = -k A_c \frac{dT}{dx} \), and letting \( \theta = T - T_\infty \):

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

Boundary Conditions

• At base (\( x=0 \)): \( \theta = \theta_b \)

• At tip (\( x=L \)):

  • Infinite fin: \( \theta(L) = 0 \)

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

  • Insulated tip: \( \frac{d\theta}{dx}\big|_{x=L} = 0 \)

Solutions and Heat Transfer Rate

• Insulated tip:

$$ \theta(x) = \theta_b \frac{\cosh[m(L-x)]}{\cosh(mL)}, \quad q_f = \sqrt{h P k A_c} \theta_b \tanh(mL) $$

• Infinite fin:

$$ \theta(x) = \theta_b e^{-mx}, \quad q_f = \sqrt{h P k A_c} \theta_b, \quad \eta_f = \frac{1}{mL} $$

• Convective tip:

$$ q_f = \sqrt{h P k A_c} \theta_b \frac{\sinh(mL) + (h/(mk))\cosh(mL)}{\cosh(mL) + (h/(mk))\sinh(mL)} $$

[!TIP] For short fins (\( mL < 0.5 \)), efficiency is high; for long fins, efficiency drops. Use infinite fin approximation only if \( mL > 3 \).

Fin Design Considerations

• Material: high thermal conductivity (e.g., Al, Cu).

• Geometry: thin, long, large perimeter-to-area ratio.

• Attachment: good thermal contact at base.

• Tip condition: insulated tip often assumed for simplicity.

3. Critical Insulation Thickness

Concept

For cylinders/pipes, adding insulation can increase heat loss if the pipe radius \( r_i < r_{cr} \). This occurs because insulation adds conductive resistance but also increases convective surface area.

Derivation for Cylinder

Heat loss per unit length:

$$ q = \frac{2\pi (T_i - T_\infty)}{\frac{\ln(r_o/r_i)}{k} + \frac{1}{h r_o}} $$

Set \( \frac{dq}{dr_o} = 0 \) (or minimize total resistance \( R \)):

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

Thus, critical outer radius:

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

Plot of Heat Transfer vs. Outer Radius

• If \( r_i > r_{cr} \): \( q \) decreases monotonically with \( r_o \).

• If \( r_i < r_{cr} \): \( q \) increases with \( r_o \) until \( r_o = r_{cr} \) (maximum), then decreases.

  [!NOTE] The question mentions "minimum point" (Dec 2025), but it is actually a maximum for heat loss when \( r_i < r_{cr} \).

Application in Design

• Refrigeration systems: Pipes often have small \( r_i \); insulation must be thicker than \( r_{cr} - r_i \) to avoid increased heat gain.

• Electrical cables: Insulation thickness chosen to limit heat loss or maintain temperature.

Calculation Problems

• Critical thickness: \( t_{cr} = r_{cr} - r_i \).

• Ratio of heat loss:

$$ \frac{q_{\text{with ins}}}{q_{\text{without}}} = \frac{\ln(r_o/r_i)/k + 1/(h r_i)}{\ln(r_o/r_i)/k + 1/(h r_o)} $$

(Without insulation: \( q_{\text{ns}} = 2\pi r_i h (T_i - T_\infty) \))

4. Convection Heat Transfer

Boundary Layer Theory

• Hydrodynamic boundary layer: Region where velocity changes from 0 to \( u_\infty \).

• Thermal boundary layer: Region where temperature changes from \( T_s \) to \( T_\infty \).

• Momentum Equation (Flat Plate, Boundary Layer Approximations):

  Continuity: \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \)

  x-Momentum: \( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} \)

  (Derived from Navier-Stokes by neglecting \( \frac{\partial u}{\partial y} \ll \frac{\partial u}{\partial x} \), \( v \ll u \), and pressure gradient across layer.)

• Displacement thickness \( \delta^* \):

$$ \delta^* = \int_0^\infty \left(1 - \frac{u}{u_\infty}\right) dy $$

Represents distance outer stream is displaced due to boundary layer.

• Momentum thickness \( \theta \):

$$ \theta = \int_0^\infty \frac{u}{u_\infty} \left(1 - \frac{u}{u_\infty}\right) dy $$

Related to momentum deficit.

Dimensionless Numbers

Flow Regimes

• Laminar: Smooth, ordered (Re < 5×10⁵ for flat plate).

• Transition: 5×10⁵ < Re < 3×10⁶.

• Turbulent: Chaotic, mixing (Re > 3×10⁶); higher \( h \).

Forced Convection

• External Flow:

  • Flat Plate:

    Laminar (local): \( Nu_x = 0.332 Re_x^{1/2} Pr^{1/3} \) (Pr > 0.6)

    Laminar (average): \( \bar{Nu}_L = 0.664 Re_L^{1/2} Pr^{1/3} \)

    Turbulent (local): \( Nu_x = 0.0296 Re_x^{4/5} Pr^{1/3} \)

  • Cylinder (Cross-Flow):

    Use \( \bar{Nu}_D = C Re_D^m Pr^{1/3} \) (C, m from tables) or Churchill-Bernstein for all Re:

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

• Internal Flow (Circular Tube):

  • Hydrodynamically Developed: Velocity profile fully developed (Blasius for laminar, 1/7 power for turbulent).

  • Thermally Developed: Temperature profile fully developed.

  • Correlations:

    Laminar, fully developed: \( Nu = 3.66 \) (constant \( T_s \)), \( Nu = 4.36 \) (constant \( q'' \)).

    Turbulent (Dittus-Boelter): \( Nu = 0.023 Re^{0.8} Pr^{0.4} \) (heating), \( Pr^{0.3} \) (cooling).

    Sieder-Tate (variable properties): \( Nu = 0.027 Re^{0.8} Pr^{1/3} (\mu/\mu_w)^{0.14} \).

  • Mass Average Velocity:

$$ u_{ma} = \frac{\int \rho u^2 dA}{\int \rho u dA} $$

Used for variable properties to define bulk temperature.

Natural Convection

• Vertical Plate:

  Laminar (\( Ra_L < 10^9 \)): \( \bar{Nu} = 0.59 Ra_L^{1/4} \)

  Turbulent (\( Ra_L > 10^9 \)): \( \bar{Nu} = 0.10 Ra_L^{1/3} \)

• Horizontal Cylinder:

  Use Churchill-Chu correlation:

$$ \bar{Nu} = \left[0.60 + \frac{0.387 Ra_D^{1/6}}{[1+(0.559/Pr)^{9/16}]^{8/27}}\right]^2 $$

Boiling and Condensation

• Pool Boiling Regimes: Natural convection → nucleate boiling (high \( h \)) → critical heat flux → transition boiling → film boiling (low \( h \), Leidenfrost effect).

• Flow Boiling: Boiling in flowing fluid (e.g., in tubes).

• Condensation:

  • Filmwise: Continuous film, lower \( h \).

  • Dropwise: Droplets, higher \( h \) (2–5×), but requires special surface.

5. Radiation Heat Transfer

Fundamental Properties

• Emissivity \( \epsilon \): Ratio of radiation from surface to black body at same \( T \).

• Absorptivity \( \alpha \): Fraction of incident radiation absorbed.

• Reflectivity \( \rho \): Fraction reflected.

• Transmissivity \( \tau \): Fraction transmitted (opaque: \( \tau = 0 \)).

• Kirchhoff’s Law: For a surface in thermal equilibrium, \( \alpha = \epsilon \) at given wavelength and direction. For diffuse gray surfaces, \( \alpha = \epsilon \) (total hemispherical).

Black Body Radiation

• Planck’s Law (Spectral):

$$ E_{\lambda b} = \frac{C_1}{\lambda^5 (e^{C_2/(\lambda T)} - 1)} $$

\( C_1 = 3.742 \times 10^8 \, \text{W} \mu\text{m}^4/\text{m}^2 \), \( C_2 = 1.439 \times 10^4 \, \mu\text{m} \cdot \text{K} \).

• Wien’s Displacement Law:

$$ \lambda_{\text{max}} T = b \quad (b = 2898 \, \mu\text{m·K}) $$

• Stefan-Boltzmann Law:

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

Radiation Shape Factors (View Factors)

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

• Properties:

  • Reciprocity: \( A_i F_{i\to j} = A_j F_{j\to i} \)

  • Summation: \( \sum_{j=1}^N F_{i\to j} = 1 \)

  • Symmetry: \( F_{i\to j} = F_{j\to i} \) only if \( A_i = A_j \).

• Common Geometries:

  | Geometry | \( F_{1\to2} \) | |----------|----------------| | Parallel plates (equal area \( A \), separation \( L \ll \sqrt{A} \)) | ≈ 1 | | Concentric cylinders (inner to outer) | 1 | | Hemisphere to base | \( F_{\text{hem}\to\text{base}} = 1/2 \) | | Cylinder base to curved surface | \( F_{\text{base}\to\text{curved}} = 1 \) |

• Derivation for Two Surfaces of Equal Area:

  From reciprocity \( A_1 F_{1\to2} = A_2 F_{2\to1} \) and since \( A_1 = A_2 \), \( F_{1\to2} = F_{2\to1} \).

  From summation for a two-surface enclosure: \( F_{1\to1} + F_{1\to2} = 1 \), \( F_{2\to2} + F_{2\to1} = 1 \).

  If both surfaces are convex (no self-view), \( F_{1\to1} = F_{2\to2} = 0 \), so \( F_{1\to2} = F_{2\to1} = 1 \).

Radiation Between Diffuse Gray Surfaces

• Radiosity \( J \): Total radiation leaving a surface per unit area:

$$ J = \epsilon E_b + (1-\epsilon) G $$

where \( G \) is irradiation (total incident radiation).

• Irradiation \( G \):

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

• Net Heat Exchange (Two Surfaces):

$$ q_{1\to2} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1-\epsilon_1}{A_1 \epsilon_1} + \frac{1}{A_1 F_{1\to2}} + \frac{1-\epsilon_2}{A_2 \epsilon_2}} $$

• Reciprocity Theorem: \( A_i F_{i\to j} = A_j F_{j\to i} \) (already stated).

Radiation Shields

• Purpose: Reduce net radiation by adding low-emissivity surfaces.

• Effectiveness: Each shield approximately halves heat transfer if emissivities equal.

Numerical Problems

• Net radiation between two plates (Nov 2023): Use three-surface enclosure (plates + room) with radiosity method.

• Cylindrical enclosure (May 2024): For black concentric cylinders, \( q = \frac{2\pi L \sigma (T_1^4 - T_2^4)}{\frac{1}{D_1} - \frac{1}{D_2}} \).

• Hemispherical cavity (Dec 2025):

  For a cavity, net radiation to surroundings at \( T_{\text{sur}} \):

$$ q = \frac{\sigma (T_s^4 - T_{\text{sur}}^4)}{\frac{1-\epsilon}{A_s \epsilon} + \frac{1}{A_s F_{s\to\text{op}}}} $$

For hemisphere: \( A_s = 2\pi R^2 \), \( F_{s\to\text{op}} = 0.5 \). If surroundings are large and black, \( T_{\text{op}} = T_{\text{sur}} \).

6. Heat Exchangers

Classification

• By Flow Arrangement:

  • Parallel flow: Both fluids same direction.

  • Counter-flow: Opposite directions (higher LMTD, more efficient).

  • Cross-flow: Perpendicular (e.g., car radiator).

  • Cross-counterflow: Combination.

• By Construction:

  • Shell-and-tube: One fluid in tubes, other over tubes in shell.

  • Plate: Corrugated plates for high area density.

  • Air-cooled: Fins on tubes, air cooled by fans.

  • Regenerative: Same matrix alternately exposed to hot and cold fluids.

• By Phase Change: Condensers (vapor → liquid), evaporators (liquid → vapor).

Log Mean Temperature Difference (LMTD)

• Derivation for Parallel Flow:

  Energy balance: \( dq = U dA \Delta T(x) \), \( dq = C_h dT_h = -C_c dT_c \).

  For constant \( U \), \( C_h \), \( C_c \):

$$ q = U A \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)} $$

where \( \Delta T_1 = T_{h,\text{in}} - T_{c,\text{in}} \), \( \Delta T_2 = T_{h,\text{out}} - T_{c,\text{out}} \).

• For Counter-Flow:

  \( \Delta T_1 = T_{h,\text{in}} - T_{c,\text{out}} \), \( \Delta T_2 = T_{h,\text{out}} - T_{c,\text{in}} \).

• LMTD Correction Factor (F):

  For cross-flow or multi-pass, \( \Delta T_{\text{lm, actual}} = F \cdot \Delta T_{\text{lm, cf}} \).

  \( F \) obtained from charts using:

$$ P = \frac{T_{h,\text{in}} - T_{h,\text{out}}}{T_{h,\text{in}} - T_{c,\text{in}}}, \quad R = \frac{T_{h,\text{in}} - T_{c,\text{in}}}{T_{c,\text{out}} - T_{c,\text{in}}} $$

Effectiveness-NTU Method (Overview)

• Effectiveness: \( \epsilon = \frac{q}{q_{\text{max}}} \), \( q_{\text{max}} = C_{\text{min}} (T_{h,\text{in}} - T_{c,\text{in}}) \).

• NTU: \( \text{NTU} = \frac{UA}{C_{\text{min}}} \).

• Counter-Flow:

$$ \epsilon = \frac{1 - e^{-\text{NTU}(1-C_r)}}{1 - C_r e^{-\text{NTU}(1-C_r)}}, \quad C_r = \frac{C_{\text{min}}}{C_{\text{max}}} $$

Design and Performance Calculations

Given \( U \), \( A \), flow rates, inlet temperatures, find outlet temperatures:

1. Compute \( C_h \), \( C_c \), \( C_{\text{min}} \), \( C_{\text{max}} \), \( C_r \).

2. Assume \( \Delta T_{\text{lm}} \) (use LMTD or effectiveness).

3. Solve energy balance: \( q = C_h (T_{h,\text{in}} - T_{h,\text{out}}) = C_c (T_{c,\text{out}} - T_{c,\text{in}}) = U A \Delta T_{\text{lm}} \).

4. Iterate if necessary (unknown outlet temps).

Fouling

• Causes: Scale, corrosion, biological growth on surfaces.

• Effects: Increases thermal resistance, reduces \( U \), increases pumping power.

• Fouling Factor \( R_f \): Added to resistance:

$$ \frac{1}{U_{\text{actual}}} = \frac{1}{U_{\text{clean}}} + R_f $$

• Impact: Regular cleaning, material selection, water treatment.

Temperature Profiles

• Equal Slopes Condition: \( \left| \frac{dT_h}{dA} \right| = \left| \frac{dT_c}{dA} \right| \) when \( C_h = C_c \).

• Effect in Counter-Flow: Temperature profiles are parallel; LMTD equals arithmetic mean. Maximizes effectiveness for given NTU when \( C_r = 1 \).

7. Transient Heat Conduction

Lumped System Analysis

• Criteria: Biot number \( Bi = \frac{h L_c}{k} < 0.1 \), where \( L_c = V/A_s \).

• Temperature-Time Relation:

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

• Characteristic Length \( L_c \):

  | Geometry | \( L_c \) | |----------|-----------| | Plane wall (thickness \( L \)) | \( L/2 \) | | Long cylinder (radius \( r \)) | \( r/2 \) | | Sphere (radius \( r \)) | \( r/3 \) |

• Applications: Solids with high conductivity (metals) in low-\( h \) fluids.

• Numerical Problems: Check \( Bi \), then find time to reach temperature or temperature at given time.

Heisler Charts

• Graphical solutions for 1D transient in infinite slab, cylinder, sphere.

• Use dimensionless temperature \( \theta/\theta_i \) vs. Fourier number \( Fo = \alpha t / L_c^2 \).

• For slab: use half-thickness \( L \); cylinder/sphere: use radius \( r \).

Infinite Thermal Conductivity Method (Product Solution)

• For multi-dimensional transient problems when \( Bi < 0.1 \) in all directions.

• Temperature distribution approximated as product of 1D solutions:

$$ \frac{T - T_\infty}{T_i - T_\infty} = \prod_{i=1}^{3} \frac{\theta_i}{\theta_{i0}} $$

where each \( \theta_i \) is 1D solution for that coordinate.

[!TIP] Lumped capacitance is invalid for large \( Bi \); use Heisler charts or numerical methods.

8. Dimensional Analysis

Buckingham Pi Theorem

If a physical relation involves \( n \) variables and \( r \) fundamental dimensions, it can be expressed in terms of \( (n-r) \) independent dimensionless Pi terms:

$$ F(\pi_1, \pi_2, ..., \pi_{n-r}) = 0 $$

Step-by-Step Procedure

1. List all variables and their dimensions.

2. Determine \( r \) (number of fundamental dimensions, usually 3: M, L, T; sometimes Θ for temperature).

3. Select \( r \) repeating variables that include all fundamental dimensions.

4. Form Pi terms: each Pi = (non-repeating variable) × (repeating variables)^exponents. Solve for exponents to make Pi dimensionless.

5. Write functional relation.

Advantages

• Reduces number of experiments.

• Generalizes results across scales.

• Reveals dominant physical mechanisms.

Limitations

• Does not give exact functional form.

• Cannot determine dimensionless constants.

• Choice of repeating variables can affect Pi terms.

Application to Forced Convection

Variables: \( h, k, \rho, \mu, c_p, u, D \).

Dimensions: \( h: MT^{-3}\Theta^{-1} \), \( k: MLT^{-3}\Theta^{-1} \), \( \rho: ML^{-3} \), \( \mu: ML^{-1}T^{-1} \), \( c_p: L^2T^{-2}\Theta^{-1} \), \( u: LT^{-1} \), \( D: L \).

Fundamental dimensions: M, L, T, Θ → \( r=4 \), \( n=7 \) → 3 Pi terms.

Choose repeating variables: \( \rho, u, D, c_p \) (cover M, L, T, Θ).

Then:

• \( \pi_1 = \frac{h D}{k} = Nu \)

• \( \pi_2 = \frac{\rho u D}{\mu} = Re \)

• \( \pi_3 = \frac{\mu c_p}{k} = Pr \)

Thus, \( Nu = f(Re, Pr) \).

Application to Fin Performance

Variables: \( q_f, k, h, t, L, \Delta T, \rho, c_p, \mu \) (air properties).

But \( q_f \) depends on fin geometry and \( h \). Typically, key groups: \( \eta_f \) (efficiency) and \( mL \) (where \( m = \sqrt{hP/(kA_c)} \)).

From dimensional analysis, we obtain Pi terms such as \( \frac{q_f}{k \Delta T L} \), \( \frac{hL}{k} \), \( \frac{L}{t} \), \( Re \), \( Pr \). However, for steady-state fins, \( \rho, c_p, \mu \) are not directly involved unless \( h \) is expressed in terms of them.

9. Mass Transfer

Modes of Mass Transfer

• Diffusion: Due to concentration gradient (Fick’s law).

• Convection: Bulk motion + diffusion.

• Mass Transfer Through Stagnant Media: Diffusion through a non-diffusing species (e.g., evaporation through air).

Fick’s First Law

$$ J = -D \frac{dc}{dx} $$

\( J \): molar flux (mol/m²·s), \( D \): diffusion coefficient (m²/s), \( c \): molar concentration (mol/m³).

General Equation of Mass Diffusion (Stationary Medium)

From continuity: \( \frac{\partial c}{\partial t} = -\nabla \cdot \mathbf{J} \).

With Fick’s law: \( \frac{\partial c}{\partial t} = D \nabla^2 c \).

For steady state: \( \nabla^2 c = 0 \) (analogous to heat conduction).

Binary Mixtures

• Molar concentration: \( c = \frac{P}{RT} \) (ideal gas), or \( c = \frac{\rho}{M} \) (liquids).

• Mass concentration: \( \rho = \sum \rho_i \).

• Molar fraction \( x_i \):

$$ x_i = \frac{c_i}{c} = \frac{p_i}{P} \quad \text{(Dalton’s law for ideal gases)} $$

• Mass fraction \( Y_i \):

$$ Y_i = \frac{\rho_i}{\rho} = \frac{x_i M_i}{M_{\text{avg}}}, \quad M_{\text{avg}} = \sum x_i M_i $$

• Calculations (Given Partial Pressures \( p_i \), Total \( P \), Temperature \( T \)):

  1. Molar concentration: \( c = P/(RT) \).

  2. Molar fractions: \( x_i = p_i/P \).

  3. Average molar mass: \( M_{\text{avg}} = \sum x_i M_i \).

  4. Mass density of mixture: \( \rho = P M_{\text{avg}}/(RT) \).

  5. Mass density of species \( i \): \( \rho_i = c M_i = (p_i/(RT)) M_i \).

  6. Mass fractions: \( Y_i = \rho_i/\rho = (p_i M_i)/(P M_{\text{avg}}) \).

Example (May 2024):

O₂/N₂ mixture at 15°C (288 K), \( P = 1.1 \) bar, \( p_{\text{O}_2} : p_{\text{N}_2} = 0.21 : 0.79 \).

\( p_{\text{O}_2} = 0.231 \) bar, \( p_{\text{N}_2} = 0.869 \) bar.

\( M_{\text{O}_2} = 32 \) g/mol, \( M_{\text{N}_2} = 28 \) g/mol.

\( M_{\text{avg}} = 0.21 \times 32 + 0.79 \times 28 = 28.84 \) g/mol = 0.02884 kg/mol.

\( c = P/(RT) = 1.1 \times 10^5 / (8.314 \times 288) = 45.9 \) mol/m³.

\( \rho = P M_{\text{avg}}/(RT) = 1.1 \times 10^5 \times 0.02884 / (8.314 \times 288) = 1.32 \) kg/m³.

\( Y_{\text{O}_2} = (p_{\text{O}_2} M_{\text{O}_2})/(P M_{\text{avg}}) = (0.231 \times 0.032)/(1.1 \times 0.02884) = 0.233 \), \( Y_{\text{N}_2} = 0.767 \).

Convective Mass Transfer

• Mass Transfer Coefficient \( h_m \):

$$ N_A = h_m (c_{A,s} - c_{A,\infty}) \quad \text{or} \quad N_A = h_m \rho (Y_{A,s} - Y_{A,\infty}) $$

for gases.

• Analogy with Heat Transfer:

  Reynolds \( Re = \frac{\rho u L}{\mu} \), Schmidt \( Sc = \frac{\mu}{\rho D} \), Sherwood \( Sh = \frac{h_m L}{D} \).

  Similar to Nu = f(Re, Pr).

  Example (flat plate): \( Sh_x = 0.332 Re_x^{1/2} Sc^{1/3} \).

• Applications: Evaporation, drying, absorption, distillation.

10. Additional Topics and Devices

Thermocouples

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

• Construction: Two wires (e.g., Chromel-Alumel for Type K) joined at measuring junction; other ends connected to voltmeter at reference (cold) junction.

• Applications: Temperature measurement in furnaces, engines, exhausts, industrial processes.

• Advantages: Wide range (-200°C to 2300°C), rugged, inexpensive, no external power, fast response.

• Limitations: Requires cold junction compensation, nonlinear output, drift over time, lower accuracy than RTDs, small output (μV/°C).

[!TIP] Common types: Type K (Ni-Cr/Ni-Al), Type J (Fe-CuNi), Type T (Cu-CuNi). Use extension wires to avoid errors.

End of Unit 5 Notes
Always cross-check formulas with RGPV syllabus and past papers. Practice numerical problems from Dec 2025, Jun 2025, May 2024, May 2023, Nov 2023 for exam readiness.