GATE Metallurgical Engineering Transport Phenomena and Rate Processes Syllabus

Define thermal resistance for conduction through a plane wall.

$$R_{cond} = \frac{L}{k A}$$ analogous to electrical resistance, with heat rate $Q = \frac{\Delta T}{R_{cond}}$ playing the role of current and $\Delta T$ the potential difference.

Give the steady-state radial conduction heat rate through a hollow cylinder.

$$Q = \frac{2 \pi k L (T_1 - T_2)}{\ln(r_2 / r_1)}$$ where $r_1, r_2$ are the inner and outer radii and $L$ is the cylinder length.

Give the steady-state radial conduction heat rate through a hollow sphere.

$$Q = \frac{4 \pi k (T_1 - T_2)}{\frac{1}{r_1} - \frac{1}{r_2}}$$ where $r_1$ and $r_2$ are the inner and outer radii.

How do thermal resistances combine for a composite plane wall in series?

They add: $$R_{total} = \sum_i \frac{L_i}{k_i A}$$ and the overall heat rate is $Q = \frac{\Delta T_{overall}}{R_{total}}$. This is analogous to resistors in series carrying the same current.

What is convection and what are its two types?

Convection is heat transfer between a surface and a moving fluid, combining conduction and bulk fluid motion. The two types are natural (free) convection, driven by buoyancy from density differences, and forced convection, driven by an external means (pump, fan).

State Newton's law of cooling for convection.

$$Q = h A (T_s - T_\infty)$$ where $h$ is the convective heat transfer coefficient ($\text{W·m}^{-2}\text{K}^{-1}$), $A$ the surface area, $T_s$ the surface temperature, and $T_\infty$ the bulk fluid temperature.

Define the Nusselt number and state its physical meaning.

$$Nu = \frac{h L}{k}$$ It is the ratio of convective to conductive heat transfer across a fluid layer; $Nu = 1$ implies pure conduction, while larger values indicate stronger convection.

Give a common correlation for the Nusselt number in turbulent forced convection inside a pipe.

The Dittus–Boelter equation: $$Nu = 0.023\, Re^{0.8} Pr^{n}$$ with $n = 0.4$ for heating and $n = 0.3$ for cooling, valid for turbulent flow ($Re > 10^4$).

On which dimensionless numbers does the forced-convection heat transfer coefficient depend?

Forced convection is correlated as $Nu = f(Re, Pr)$ — the Nusselt number depends on the Reynolds number (flow regime) and the Prandtl number (fluid momentum-to-thermal diffusivity ratio). Natural convection instead uses $Nu = f(Gr, Pr)$.

What is radiation as a mode of heat transfer?

Radiation is heat transfer by electromagnetic waves (no medium required). All bodies above absolute zero emit thermal radiation, and the rate depends strongly on absolute temperature ($\propto T^{4}$) and surface emissivity.

Define a black body.

A black body is an idealized surface that absorbs all incident radiation (absorptivity $\alpha = 1$) at every wavelength and angle, and is also the perfect emitter, radiating the maximum possible energy at a given temperature (emissivity $\varepsilon = 1$).

State the Stefan–Boltzmann law for a black body.

$$E_b = \sigma T^{4}$$ where $E_b$ is the total emissive power ($\text{W·m}^{-2}$), $T$ is absolute temperature, and $\sigma = 5.67 \times 10^{-8}\ \text{W·m}^{-2}\text{K}^{-4}$ is the Stefan–Boltzmann constant.

How is the Stefan–Boltzmann law modified for a real (gray) body?

$$E = \varepsilon \sigma T^{4}$$ where $\varepsilon$ (the emissivity, $0 \le \varepsilon \le 1$) accounts for the surface emitting less than an ideal black body at the same temperature.

State Kirchhoff's law of thermal radiation.

At thermal equilibrium, a body's emissivity equals its absorptivity: $\varepsilon = \alpha$. A good absorber is an equally good emitter at the same temperature and wavelength.

Define emissivity, absorptivity, reflectivity, and transmissivity and their sum.

Emissivity $\varepsilon$ is the ratio of a surface's emission to a black body's. For incident radiation, absorptivity $\alpha$, reflectivity $\rho$, and transmissivity $\tau$ are the absorbed, reflected, and transmitted fractions, and they satisfy $$\alpha + \rho + \tau = 1.$$

Write the net radiation heat exchange between two large parallel black plates.

$$Q = \sigma A (T_1^{4} - T_2^{4})$$ where $A$ is the area and $T_1, T_2$ are the absolute temperatures of the two surfaces. For gray surfaces, multiply by an emissivity factor.

State Wien's displacement law.

$$\lambda_{max} T = 2.898 \times 10^{-3}\ \text{m·K}$$ The wavelength of peak black-body emission is inversely proportional to absolute temperature; hotter bodies radiate at shorter wavelengths.

State Fick's first law of diffusion.

$$J_A = -D_{AB} \frac{dC_A}{dx}$$ where $J_A$ is the molar flux of species A ($\text{mol·m}^{-2}\text{s}^{-1}$), $D_{AB}$ is the diffusion coefficient ($\text{m}^{2}\text{s}^{-1}$), and $\frac{dC_A}{dx}$ is the concentration gradient. Mass flows down the concentration gradient.

State Fick's second law of diffusion (unsteady-state).

$$\frac{\partial C_A}{\partial t} = D_{AB} \frac{\partial^{2} C_A}{\partial x^{2}}$$ It describes how the concentration of a diffusing species changes with time in one dimension for constant $D_{AB}$.

What is the physical meaning of the diffusion coefficient and how does it vary for gases, liquids, solids?

The diffusion coefficient $D_{AB}$ measures how fast species A spreads through B (units $\text{m}^{2}\text{s}^{-1}$). Typical magnitudes: gases $\sim 10^{-5}$, liquids $\sim 10^{-9}$, solids $\sim 10^{-12}$ or smaller; it increases with temperature.

Define the convective mass transfer coefficient and write the mass flux relation.

$$N_A = k_c (C_{A,s} - C_{A,\infty})$$ where $k_c$ is the convective mass transfer coefficient ($\text{m·s}^{-1}$), and $C_{A,s}, C_{A,\infty}$ are the surface and bulk concentrations of A. It is the mass-transfer analogue of Newton's law of cooling.

Define the Sherwood number and give its analogy in heat transfer.

$$Sh = \frac{k_c L}{D_{AB}}$$ It is the ratio of convective to diffusive mass transport, and is the mass-transfer analogue of the Nusselt number $Nu = \frac{hL}{k}$.

State the Buckingham Pi theorem.

If a physical problem involves $n$ variables expressible in $m$ fundamental dimensions, the relationship can be reduced to $(n - m)$ independent dimensionless groups (Pi terms). It is the basis of dimensional analysis.

In the Buckingham Pi theorem, how do you choose repeating variables?

Select $m$ repeating variables (equal to the number of fundamental dimensions) that together contain all the dimensions, are dimensionally independent, and do not by themselves form a dimensionless group. The dependent variable should not be chosen as a repeating variable.

Define the Reynolds number and state its physical significance.

$$Re = \frac{\rho v L}{\mu} = \frac{v L}{\nu}$$ It is the ratio of inertial to viscous forces and characterizes the flow regime (laminar vs. turbulent).

Define the Prandtl number and state its significance.

$$Pr = \frac{\mu c_p}{k} = \frac{\nu}{\alpha}$$ It is the ratio of momentum diffusivity to thermal diffusivity, relating the relative thickness of the velocity and thermal boundary layers.

Define the Schmidt number and state its significance.

$$Sc = \frac{\mu}{\rho D_{AB}} = \frac{\nu}{D_{AB}}$$ It is the ratio of momentum diffusivity to mass diffusivity, the mass-transfer analogue of the Prandtl number, relating velocity and concentration boundary layers.

Define the Grashof number and state where it is used.

$$Gr = \frac{g \beta (T_s - T_\infty) L^{3}}{\nu^{2}}$$ It is the ratio of buoyancy to viscous forces and governs natural (free) convection; it plays the role that $Re$ does in forced convection.

Define the Biot number and the Nusselt number and contrast them.

$$Bi = \frac{h L}{k_{solid}}, \qquad Nu = \frac{h L}{k_{fluid}}$$ The Biot number uses the solid's conductivity to compare internal conduction with surface convection (key for lumped-capacitance analysis), whereas the Nusselt number uses the fluid's conductivity to characterize convective heat transfer.

State the Chilton–Colburn analogy linking momentum, heat, and mass transfer.

$$\frac{f}{2} = j_H = j_M, \quad \text{with } j_H = St\, Pr^{2/3}, \ j_M = St_m\, Sc^{2/3}$$ It relates the friction factor to the heat ($j_H$) and mass ($j_M$) transfer factors, enabling estimation of $h$ or $k_c$ from friction data.