GATE Chemical Engineering Heat Transfer Syllabus

State the Reynolds analogy and the Chilton–Colburn (modified Reynolds) analogy.

Reynolds analogy: $St = \frac{f}{2}$ (valid for $Pr = 1$). Chilton–Colburn: $$St\,Pr^{2/3} = \frac{f}{2}$$ i.e. $j_H = St\,Pr^{2/3} = \frac{f}{2}$, valid for $0.6 < Pr < 60$.

Define the overall heat transfer coefficient $U$ for a plane wall in terms of individual resistances.

$$\frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}$$ so $q = UA\,\Delta T$. $U$ combines both convective films and the conductive wall into one coefficient.

Write the overall heat transfer coefficient based on outside area for a tube, including fouling.

$$\frac{1}{U_o A_o} = \frac{1}{h_i A_i} + \frac{R_{f,i}}{A_i} + \frac{\ln(r_o/r_i)}{2\pi k L} + \frac{R_{f,o}}{A_o} + \frac{1}{h_o A_o}$$ where $R_f$ are the fouling (dirt) resistances.

What is a fouling factor (dirt factor) and how does it affect $U$?

Fouling factor $R_f$ ($\mathrm{m^2\cdot K/W}$) is the added thermal resistance from deposits/scale on heat-transfer surfaces. It increases total resistance, lowering $U$ and reducing exchanger performance over time.

Define the Stefan–Boltzmann law for a black body.

$$E_b = \sigma T^4$$ where $E_b$ is the blackbody emissive power ($\mathrm{W/m^2}$), $\sigma = 5.67\times10^{-8}\ \mathrm{W/m^2\cdot K^4}$, and $T$ is absolute temperature.

Define emissivity, and state Kirchhoff's law of radiation.

Emissivity $\varepsilon = \frac{E}{E_b}$ is the ratio of a body's emissive power to a blackbody's at the same temperature ($0 \le \varepsilon \le 1$). Kirchhoff's law: at thermal equilibrium, emissivity equals absorptivity, $\varepsilon = \alpha$.

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

$$q = \frac{\sigma A (T_1^4 - T_2^4)}{\dfrac{1}{\varepsilon_1} + \dfrac{1}{\varepsilon_2} - 1}$$ The denominator term is the combined surface resistance factor for the two gray surfaces.

Define a view (shape/configuration) factor $F_{12}$ and state the reciprocity relation.

$F_{12}$ is the fraction of radiation leaving surface 1 that directly reaches surface 2. Reciprocity: $A_1 F_{12} = A_2 F_{21}$. Summation rule: $\sum_j F_{ij} = 1$ for an enclosure.

State Wien's displacement law and Planck's distribution context.

Wien's law: the wavelength of maximum emission shifts with temperature as $$\lambda_{max} T = 2898\ \mu\mathrm{m\cdot K}$$ It is the peak of Planck's spectral blackbody distribution; hotter bodies emit at shorter wavelengths.

For an opaque surface, relate absorptivity, reflectivity, and transmissivity; and for a gray body.

In general $\alpha + \rho + \tau = 1$. For an opaque body $\tau = 0$, so $\alpha + \rho = 1$. A gray body has $\alpha$ (and $\varepsilon$) independent of wavelength.

What is the fundamental difference between a recuperator and a regenerator?

Recuperator: hot and cold fluids are separated by a wall and exchange heat continuously through it (most common exchangers). Regenerator: fluids alternately pass over the same heat-storage matrix, which stores and releases heat cyclically.

Compare parallel-flow and counter-flow heat exchangers.

Parallel flow: both fluids enter at the same end, flow same direction; outlet temperatures approach an intermediate value, larger $\Delta T$ at inlet. Counter flow: fluids flow in opposite directions; gives larger LMTD, higher effectiveness, and cold outlet can exceed hot outlet.

Classify the main types of heat exchangers by construction.

Double-pipe (concentric tube), shell-and-tube, plate (gasketed/brazed/welded), plate-fin/compact, spiral, and extended-surface (finned) exchangers. By flow: parallel, counter, and cross-flow.

Derive/state the LMTD and its use in exchanger sizing.

$$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$$ Used as $q = U A\, F\, \Delta T_{lm}$, where $F$ is a correction factor (=1 for pure counter/parallel flow, <1 for cross/multipass).

Define heat exchanger effectiveness $\varepsilon$ and the NTU in the $\varepsilon$–NTU method.

$$\varepsilon = \frac{q}{q_{max}} = \frac{q}{C_{min}(T_{h,i} - T_{c,i})}$$ and $$NTU = \frac{UA}{C_{min}}$$ where $C = \dot{m}c_p$ and $C_{min}$ is the smaller heat-capacity rate.

Write the effectiveness relation for a counter-flow exchanger (with $C_r = C_{min}/C_{max}$).

$$\varepsilon = \frac{1 - \exp[-NTU(1 - C_r)]}{1 - C_r\exp[-NTU(1 - C_r)]}$$ For $C_r = 1$: $\varepsilon = \frac{NTU}{1 + NTU}$; for $C_r = 0$ (phase change): $\varepsilon = 1 - e^{-NTU}$.

Describe the construction and use of a double-pipe heat exchanger.

Two concentric pipes: one fluid flows in the inner pipe, the other in the annulus. Simple, cheap, good for small duties and high pressures; arranged in hairpin sections. Easily adapted to true counter-current flow.

How is the equivalent (hydraulic) diameter computed for the annulus of a double-pipe exchanger?

For heat transfer: $$D_e = \frac{D_2^2 - D_1^2}{D_1}$$ (based on heated perimeter). For pressure drop the hydraulic diameter $D_h = D_2 - D_1$, where $D_2$ is inner-shell ID and $D_1$ is inner-pipe OD.

Describe the basic construction of a shell-and-tube heat exchanger.

A bundle of tubes inside a cylindrical shell. One fluid flows through the tubes (tube side), the other across/around them in the shell (shell side). Includes tube sheets, baffles, headers; designated by passes, e.g. 1-2 (one shell, two tube passes).

What is the function of baffles in a shell-and-tube exchanger?

Baffles (commonly segmental) direct shell-side fluid across the tubes increasing turbulence and shell-side $h$, increase fluid velocity and path length, and physically support the tube bundle to prevent sagging and vibration.

Compare fixed tube-sheet, U-tube, and floating-head shell-and-tube designs.

Fixed tube-sheet: cheapest, but no allowance for thermal expansion and tube exterior hard to clean. U-tube: allows expansion, only one tube sheet, but tubes hard to clean inside. Floating-head: one tube sheet floats to absorb expansion and allows bundle removal for cleaning.

What guidelines decide which fluid goes on the tube side vs. shell side?

Tube side generally takes: the corrosive, high-pressure, fouling, or lower-flow fluid (easier to clean tubes, contain pressure). Shell side suits: the fluid needing lower pressure drop, viscous fluids, or the one requiring higher heat-transfer area via baffling.

Define the function and basic operating principle of an evaporator.

An evaporator concentrates a solution by boiling off (vaporizing) the volatile solvent (usually water) using steam as the heating medium, leaving a more concentrated liquid product. It is a key unit in concentration of solutions.

List the main types of evaporators by construction.

Open-pan/jacketed, horizontal-tube, short-tube (calandria/standard vertical), long-tube vertical (rising-film and falling-film), forced-circulation, and agitated/scraped-surface (wiped-film) evaporators.

Contrast rising-film (climbing-film) and falling-film evaporators.

Rising-film: liquid boils and vapor forces liquid up the tubes as a film; needs sufficient $\Delta T$. Falling-film: liquid flows down tube walls as a thin film by gravity; low residence time and low $\Delta T$, ideal for heat-sensitive/viscous liquids.

When is a forced-circulation evaporator preferred?

For viscous solutions, those that foul or scale, or that form crystals/salt — a pump forces high-velocity liquid through tubes (suppressing boiling inside), giving high heat transfer coefficients and reduced fouling, at the cost of pumping power.

Define capacity and economy of an evaporator.

Capacity = kg of water evaporated per hour. Economy (steam economy) = kg of water evaporated per kg of steam supplied. For a single effect, economy is < 1 (~0.8–0.9); it rises roughly with the number of effects.

What is boiling point elevation (BPE) and its effect on evaporator driving force?

BPE is the increase in boiling point of a solution above that of pure solvent at the same pressure (due to dissolved solute, per Dühring's rule). It reduces the effective temperature driving force $\Delta T$ available for heat transfer, lowering capacity.

Write the single-effect evaporator steady-state mass and enthalpy balances.

Mass: $F = L + V$ and solute balance $F x_F = L x_L$. Enthalpy: $$F h_F + S\lambda_S = V H_V + L h_L$$ where $S\lambda_S$ is steam heat supplied, $V$ is vapor, $L$ is concentrated liquor. Heat duty $q = U A\,\Delta T = S\lambda_S$.

What is a multiple-effect evaporator and what is its key advantage?

Several evaporator bodies in series where vapor from one effect serves as the heating steam for the next (run at progressively lower pressures). Advantage: dramatically improved steam economy — roughly $N$ kg water evaporated per kg live steam for $N$ effects.

Compare forward-feed and backward-feed arrangements in multiple-effect evaporators.

Forward feed: liquor and steam flow same direction (1→N), no liquor pumps needed but cold feed reduces first-effect economy; concentrated liquor at lowest temperature. Backward feed: liquor enters last effect and moves counter to vapor (N→1); needs pumps but gives better economy and handles viscous concentrates (hottest where most concentrated).