GATE ME Heat Transfer Formula Sheet 2027

Heat transfer in GATE ME is steady and calculation-heavy. Conduction, convection, radiation and heat exchangers each drew between 9 and 12 marks across the six papers from 2022 to 2026, so none of them can be dropped. This sheet gives the formulas each topic runs on, a worked example for each, and the trap behind it.

Key takeaways

• The four big topics carry similar marks, so learn all four rather than betting on one.
• Most conduction questions reduce to one equation, solved for a plane wall with the right boundary conditions.
• The lumped model needs $Bi < 0.1$ on the length $V/A_s$, not on the radius.
• Many heat exchanger questions need only the definition of effectiveness and an energy balance.
• Radiation enclosure questions start with view factors by summation and reciprocity.
• Boiling and condensation are new in the 2027 syllabus, and Heisler's charts are no longer named.

The terms this sheet uses

Thermal resistance $R$ is temperature difference divided by heat rate, so resistances in series add like electrical ones. $h$ is the convection coefficient and $k$ the conductivity. $\theta_b = T_b - T_\infty$ is the excess temperature at a fin base.

Every temperature inside $T^4$, a ratio or an exponential is absolute, in kelvin. Celsius is safe only in a difference. $\sigma = 5.67 \times 10^{-8}\ \text{W/m}^2\text{K}^4$.

Conduction and thermal resistance

Example: a wall with $L/k = 0.2$, inside $h = 10$ and outside $h = 25\ \text{W/m}^2\text{K}$ has $\sum R = 0.2 + 0.1 + 0.04 = 0.34\ \text{m}^2\text{K/W}$ per square metre. A 34 K overall difference drives $100\ \text{W/m}^2$.

Example: insulation with $k = 0.1\ \text{W/m K}$ and $h = 5$ gives $r_c = 20$ mm. A 10 mm wire loses more heat as insulation is added, up to 20 mm.

Trap: a steady profile without heat generation cannot have an interior maximum. If a question shows one, there must be a source.

Fins

Fin efficiency is $\displaystyle \frac{\tanh(mL)}{mL}$ for an insulated tip. For a convecting tip, use the corrected length $\displaystyle L + \frac{t}{2}$ for a thin plate or $\displaystyle L + \frac{D}{4}$ for a pin.

Example: a 10 mm pin with $h = 20$, $k = 200$ and $\theta_b = 80$ K has $\displaystyle m = \sqrt{\frac{4h}{kD}} = 6.32\ \text{m}^{-1}$. A very long fin passes 7.95 W; a 0.1 m fin with an insulated tip passes $7.95\tanh(0.632) = 4.45$ W, at efficiency 0.885.

Fins pay where $h$ is low, as with gases, and $k$ is high.

Unsteady conduction: the lumped model

Example: a 10 mm steel ball with $k = 40$, $\rho = 7800$, $c = 500$ in air at $h = 100$ has $L_c = 1.67$ mm and $Bi = 0.0042$. Then $\tau = 65$ s, and the excess falls to a tenth in $65\ln 10 = 150$ s.

Both unsteady conduction questions in the six counted papers were lumped-model questions.

Convection

Example: doubling the free-stream velocity raises a laminar $h$ by $\sqrt{2} = 1.41$ times and a turbulent one by $2^{0.8} = 1.74$ times. With $Re = 50\,000$ and $Pr = 0.7$, heating, Dittus–Boelter gives $Nu = 0.023 \times 5743 \times 0.867 = 114.5$.

Remember: turbulent correlations go as $Re^{0.8}$ against $Re^{0.5}$ for laminar flow. That exponent is what the syllabus line on the effect of turbulence comes down to.

The book's last-minute sheet lists every Heat Transfer formula with its condition, and its chapter traces the 29 counted questions topic by topic. Both are in the GATE ME 2027 book, with 942 questions with worked solutions and 10 full mock tests.

Boiling and condensation (new in 2027)

The pool boiling curve plots heat flux against the excess temperature $T_s - T_{sat}$. It runs through free convection, nucleate boiling (very high $h$, the useful regime), transition boiling and film boiling. Past the critical heat flux, a heat-flux-controlled surface jumps to film boiling, which is burnout. The Leidenfrost point sits at the minimum heat flux.

Dropwise condensation gives a much higher $h$ than filmwise. For a laminar film on a vertical plate, Nusselt's result is:

The heat released is $\dot{m}h_{fg}$. A horizontal tube uses a smaller constant, with the diameter in place of $L$.

Heat exchangers: LMTD and NTU

Example: hot fluid 150 to 90 °C and cold 30 to 70 °C give end differences of 80 and 60 K in counterflow, so $\displaystyle \Delta T_{lm} = \frac{20}{\ln(4/3)} = 69.5$ K. In parallel flow the ends are 120 and 20 K, giving 55.8 K. Counterflow always gives the larger LMTD.

Example: a condenser with $NTU = 2$ has $\varepsilon = 1 - e^{-2} = 0.865$; a balanced counterflow unit with $NTU = 2$ has $\varepsilon = 0.667$.

Radiation

Example: plates at 800 K and 400 K with $\varepsilon = 0.5$ each exchange $\displaystyle \frac{5.67 \times 10^{-8} \times 3.84 \times 10^{11}}{3} = 7.26\ \text{kW/m}^2$. One shield of the same emissivity halves it to $3.63\ \text{kW/m}^2$. Wien puts the peak of a 1000 K black body at $2.898\ \mu\text{m}$.

Example: for long concentric cylinders of radii 0.1 and 0.4 m, $F_{12} = 1$, so reciprocity gives $F_{21} = 0.25$ and summation $F_{22} = 0.75$.

In one line: find every view factor by summation and reciprocity first, then build the network.

Using this sheet in the exam

Write the radiation network resistances and the fin formula on the rough sheet early; the exam-day rules guide explains what is provided. Logarithms, $\tanh$ and fourth powers all run on the GATE virtual calculator, so keep full precision until the last step. Many questions here are NAT with no negative marking, as the marking scheme, common to every GATE paper, explains.

More formula sheets: all of GATE ME · Fluid Mechanics · Strength of Materials · Thermodynamics

Quick revision

1. Resistances: $\displaystyle \frac{L}{kA}$, $\displaystyle \frac{\ln(r_2/r_1)}{2\pi kL}$, $\displaystyle \frac{1}{hA}$; series add.
2. Critical radius $\displaystyle \frac{k}{h}$ for a cylinder, $\displaystyle \frac{2k}{h}$ for a sphere.
3. Fin: $\displaystyle m = \sqrt{\frac{hP}{kA_c}}$; insulated tip multiplies by $\tanh(mL)$.
4. Lumped only if $Bi < 0.1$ on $L_c = V/A_s$.
5. Dittus–Boelter exponent 0.4 heating, 0.3 cooling; laminar pipe 3.66 or 4.36.
6. $\varepsilon = 1 - e^{-NTU}$ when one stream changes phase.
7. Parallel plates divide by $\displaystyle \frac{1}{\varepsilon_1} + \frac{1}{\varepsilon_2} - 1$; Wien 2898 $\mu\text{m K}$.
8. Boiling: nucleate is useful, beyond the critical heat flux lies burnout.