A small Biot number (Bi = h*Lc/k_solid, well under 0.1) justifies which simplification for an unsteady conduction problem?

The lumped-capacitance model — the solid's temperature is treated as uniform at any instant

Treating the solid as a black body radiator

Assuming steady state regardless of time

A Prandtl number much greater than 1 (Pr = momentum diffusivity / thermal diffusivity) means

the thermal boundary layer is thinner than the velocity boundary layer

the thermal boundary layer is thicker than the velocity boundary layer

the thermal boundary layer is thinner than the velocity boundary layer

conduction dominates over convection everywhere in the fluid

radiation becomes the dominant heat transfer mode

Compared to filmwise condensation, dropwise condensation on the same vertical surface gives

a much higher heat transfer coefficient, because more bare surface is exposed to incoming vapour

a much lower heat transfer coefficient, because droplets trap more vapour

a much higher heat transfer coefficient, because more bare surface is exposed to incoming vapour

exactly the same coefficient, since the same latent heat is released either way

Which of the following are true about counter-current versus co-current flow in a heat exchanger with the same terminal temperatures?

A higher LMTD means less area is needed for the same duty.

Counter-current flow is preferred whenever the process allows it.

A higher LMTD means less area is needed for the same duty.

Co-current flow can raise the cold outlet temperature above the hot outlet temperature.

Counter-current flow is preferred whenever the process allows it.

Baffles on the shell side of a shell-and-tube heat exchanger primarily

reduce the shell-side heat transfer coefficient to protect the tubes

eliminate the need for tube-side flow entirely

convert the exchanger from counter-current to co-current

Compared to a single-effect evaporator, a multiple-effect evaporator processing the same feed and duty

has a higher economy, at the cost of more total heat transfer area

has a lower economy but needs less total heat transfer area

has a higher economy, at the cost of more total heat transfer area

always uses less live steam and less total area both

For the same fluid at the same velocity, a large Biot number combined with a small Prandtl number would together suggest

internal conduction resistance in the solid cannot be neglected, while the thermal boundary layer in the fluid runs thicker than the velocity boundary layer

the lumped-capacitance model applies and the thermal boundary layer is thicker than the velocity boundary layer

internal conduction resistance in the solid cannot be neglected, while the thermal boundary layer in the fluid runs thicker than the velocity boundary layer

both quantities describe the same physical ratio and cannot differ from each other

radiation must dominate over conduction and convection

Conduction, Convection, Radiation and Heat Exchanger Design

Section 5 of the GATE Chemical Engineering (CH) paper. A process that changes temperature is moving heat by one of three mechanisms, and this section asks how fast: conduction and convection in steady and unsteady states, the boundary layer that decides a convective heat transfer coefficient, radiation as the mechanism that needs no medium at all, then boiling, condensation and evaporation as phase-change heat transfer with their own rules, and finally the two pieces of hardware every plant runs on — the heat exchanger and the evaporator — sized from those same coefficients.

1. The equation of energy and steady/unsteady conduction

The equation of energy is the differential energy balance written for a moving fluid, the thermal counterpart of the equation of motion: it tracks temperature as a function of position and time, accounting for conduction (Fourier's law, q = -kdT/dx), convection by bulk fluid motion, and any internal generation. For pure conduction with no flow, it reduces to the heat equation, dT/dt = alpha(d^2T/dx^2), where alpha = k/(rho*Cp) is thermal diffusivity — the rate at which a temperature disturbance spreads through a material, exactly analogous to momentum diffusivity (kinematic viscosity) in fluid flow.

🧠 The Biot number decides which conduction model to use

Biot number, Bi = h*Lc/k_solid (h the surface convective coefficient, Lc a characteristic length, k_solid the solid's own conductivity), compares internal conduction resistance to surface convection resistance. A small Bi (well under 0.1) means the solid's internal resistance is negligible, so its temperature is nearly uniform at any instant and the lumped-capacitance model applies; a large Bi means internal gradients matter and the full unsteady conduction equation (with its Fourier-number-dependent charts or series solution) is needed instead.

2. Convection, thermal boundary layer, and radiation

Convective heat transfer, whether natural (buoyancy-driven) or forced (pump- or fan-driven), is quantified by Newton's law of cooling, q = hA(Ts - Tinf), which simply DEFINES the heat transfer coefficient h rather than derives it. The thermal boundary layer, the region near a surface where temperature actually varies, is analogous to the velocity boundary layer, and its relative thickness against the velocity boundary layer is set by the Prandtl number (Pr = momentum diffusivity / thermal diffusivity): Pr much greater than 1 means the thermal layer is thin compared to the velocity layer, Pr much less than 1 means the opposite.

Convection versus radiation: what each mechanism needs
Mechanism	Needs a medium?	Governing relation
Conduction	Yes, a solid or stationary fluid	Fourier's law, q = -k dT/dx
Convection	Yes, a moving fluid	Newton's law of cooling, q = hA(Ts - Tinf)
Radiation	No — travels through vacuum	Stefan-Boltzmann law, q = epsilonsigmaA*T^4

Radiative heat transfer between two surfaces depends on each surface's emissivity (how close it is to an ideal black-body radiator, epsilon = 1) and their geometric view factor (what fraction of one surface's radiation actually reaches the other); because emitted power scales as absolute temperature to the FOURTH power, radiation is negligible against convection at moderate temperatures but dominates at furnace-scale temperatures, which is exactly why it is treated separately rather than folded into a single combined coefficient at low temperature.

3. Boiling, condensation and evaporation

Boiling heat transfer as a function of the surface-to-fluid temperature excess (the boiling curve) passes through natural convection, then nucleate boiling (the regime with by far the highest heat transfer coefficient, because bubbles forming and detaching agitate the liquid violently), then the critical heat flux at burnout point, then an unstable transition regime, and finally film boiling, where a stable vapour blanket insulates the surface and the coefficient drops sharply. Condensation on a vertical surface is either filmwise (the liquid wets the surface and forms a continuous film whose thickness, and therefore whose resistance, grows down the surface) or dropwise (the liquid beads up instead, exposing more bare surface and giving a MUCH higher coefficient) — dropwise is rare in practice because most surfaces are wetted.

⚠️ Do not push heat flux past the critical point on purpose

Because the boiling curve is heat-flux controlled in most real equipment (an electrical heater or a fixed process duty, not a fixed surface temperature), driving the flux just past the critical point does not move smoothly into transition boiling — it jumps discontinuously to the much lower film-boiling branch at a much higher surface temperature, often high enough to damage the heating surface. This is exactly why the critical heat flux is treated as a hard design ceiling, not merely a point of reduced efficiency.

4. Heat exchanger and evaporator types, calculations and design

A heat exchanger's duty, Q = UAdelta-T-lm (LMTD method) or Q = epsilonCmin(Th,in - Tc,in) (effectiveness-NTU method), links overall heat transfer coefficient U, area A and driving temperature difference to the actual duty; counter-current flow gives a higher LMTD than co-current for the same terminal temperatures, which is why it is preferred whenever the process allows it. A double-pipe exchanger is the simplest configuration, one pipe inside another, good for small duties or high-pressure, low-flow service; a shell-and-tube exchanger scales to industrial duties by putting many tubes inside one shell, with baffles that force the shell-side fluid to cross the tube bundle repeatedly, raising its heat transfer coefficient at the cost of extra pressure drop.

A single-effect evaporator boils off solvent using live steam once, its economy (kg vapour produced per kg steam used) capped near 1 by definition; a multiple-effect evaporator reuses the vapour from one effect as the heating medium for the next, cascading through successively lower pressures and temperatures, so its economy can exceed 1 and rises roughly with the number of effects — at the cost of a larger total heat transfer area for the same overall duty.
Design of any of these — a double-pipe or shell-and-tube exchanger, a single- or multiple-effect evaporator — ultimately solves the same equation, duty equals overall coefficient times area times driving temperature difference, for the one unknown (usually area) the calculation is asked to find, once the fluid properties and flow arrangement fix U and delta-T.

Key takeaways

The equation of energy reduces, for pure conduction with no flow, to the heat equation dT/dt = alphad2T/dx2, with thermal diffusivity alpha = k/(rhoCp) playing the same diffusive role momentum diffusivity plays in fluid flow.
A small Biot number (h*Lc/k_solid) justifies the lumped-capacitance model; a large Biot number means internal temperature gradients cannot be ignored and the full unsteady conduction solution is needed.
Newton's law of cooling defines the heat transfer coefficient h rather than deriving it; radiation, scaling with T to the fourth power, is negligible at moderate temperature but dominant at furnace scale.
Nucleate boiling gives by far the highest heat transfer coefficient on the boiling curve; pushing past the critical heat flux jumps discontinuously to the much poorer film-boiling regime, which is why it is a hard design ceiling.
A multiple-effect evaporator reuses each effect's vapour as the next effect's heating steam, pushing economy above the single-effect ceiling of about 1 at the cost of more total heat transfer area.

Practice questions (11)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

For pure conduction with no internal generation or flow, the equation of energy reduces to
1. Bernoulli's equation
2. the heat equation, dT/dt = alpha * d2T/dx2
3. the continuity equation
4. Stefan-Boltzmann's law
Show answer
Answer: B — the heat equation, dT/dt = alpha * d2T/dx2
With no bulk fluid motion and no internal heat source, the general energy equation loses its convective and generation terms and leaves exactly the diffusion (heat) equation, governed by thermal diffusivity alpha = k/(rho*Cp).
A small Biot number (Bi = h*Lc/k_solid, well under 0.1) justifies which simplification for an unsteady conduction problem?
1. The lumped-capacitance model — the solid's temperature is treated as uniform at any instant
2. Ignoring convection entirely
3. Treating the solid as a black body radiator
4. Assuming steady state regardless of time
Show answer
Answer: A — The lumped-capacitance model — the solid's temperature is treated as uniform at any instant
A small Biot number means the solid's own internal conduction resistance is negligible compared to the surface convective resistance, so temperature gradients inside the solid can be ignored and it can be treated as having one uniform temperature at each instant — exactly the lumped-capacitance assumption.
Newton's law of cooling, q = hA(Ts - Tinf),
1. derives the heat transfer coefficient h from first principles
2. defines h; its value must come from correlations, experiment or a boundary-layer solution
3. applies only to radiation, not convection
4. is valid only for natural convection, never forced convection
Show answer
Answer: B — defines h; its value must come from correlations, experiment or a boundary-layer solution
The relation is a DEFINITION of h, not a derivation of it — h itself has to be obtained separately, from a correlation (e.g. a Nusselt-number relation), an experiment, or a solved boundary-layer problem, and it applies to both natural and forced convection.
A Prandtl number much greater than 1 (Pr = momentum diffusivity / thermal diffusivity) means
1. the thermal boundary layer is thicker than the velocity boundary layer
2. the thermal boundary layer is thinner than the velocity boundary layer
3. conduction dominates over convection everywhere in the fluid
4. radiation becomes the dominant heat transfer mode
Show answer
Answer: B — the thermal boundary layer is thinner than the velocity boundary layer
Pr compares how fast momentum diffuses to how fast heat diffuses; Pr much greater than 1 means momentum diffuses faster, so the velocity boundary layer grows thicker relative to the thermal one, leaving the thermal layer comparatively thin.
On the boiling curve, the regime with the highest heat transfer coefficient is
1. natural convection
2. nucleate boiling
3. transition boiling
4. film boiling
Show answer
Answer: B — nucleate boiling
Nucleate boiling's violent bubble formation and detachment agitates the liquid near the surface far more than natural convection alone can, and film boiling's vapour blanket insulates the surface and cuts the coefficient sharply — nucleate boiling is the peak of the curve, at the critical heat flux.
Compared to filmwise condensation, dropwise condensation on the same vertical surface gives
1. a much lower heat transfer coefficient, because droplets trap more vapour
2. a much higher heat transfer coefficient, because more bare surface is exposed to incoming vapour
3. exactly the same coefficient, since the same latent heat is released either way
4. no condensation at all
Show answer
Answer: B — a much higher heat transfer coefficient, because more bare surface is exposed to incoming vapour
In dropwise condensation the liquid beads and rolls off rather than forming a continuous insulating film, so much more of the cold surface stays bare and directly exposed to incoming vapour, which is why its coefficient can run several times higher than filmwise condensation's — the reason it is prized but hard to sustain in practice.
Which of the following are true about counter-current versus co-current flow in a heat exchanger with the same terminal temperatures?
1. Counter-current flow gives a higher LMTD than co-current flow.
2. A higher LMTD means less area is needed for the same duty.
3. Co-current flow can raise the cold outlet temperature above the hot outlet temperature.
4. Counter-current flow is preferred whenever the process allows it.
Show answer
Answer: A — Counter-current flow gives a higher LMTD than co-current flow.; B — A higher LMTD means less area is needed for the same duty.; D — Counter-current flow is preferred whenever the process allows it.
The third statement is false: co-current flow's cold outlet can never exceed the hot outlet, since the two streams approach the same temperature asymptotically from a shared inlet end — it is counter-current flow, not co-current, that allows the cold outlet to exceed the hot outlet. The first, second and fourth follow directly from Q = UALMTD and counter-current's larger driving force for the same terminals.
Baffles on the shell side of a shell-and-tube heat exchanger primarily
1. reduce the shell-side heat transfer coefficient to protect the tubes
2. force the shell-side fluid to cross the tube bundle repeatedly, raising its heat transfer coefficient at the cost of extra pressure drop
3. eliminate the need for tube-side flow entirely
4. convert the exchanger from counter-current to co-current
Show answer
Answer: B — force the shell-side fluid to cross the tube bundle repeatedly, raising its heat transfer coefficient at the cost of extra pressure drop
Baffles direct the shell-side fluid in a zig-zag path across the tube bundle rather than a straight parallel run, which increases cross-flow velocity and turbulence and hence the shell-side film coefficient, at the necessary cost of a higher shell-side pressure drop.
Compared to a single-effect evaporator, a multiple-effect evaporator processing the same feed and duty
1. has a lower economy but needs less total heat transfer area
2. has a higher economy, at the cost of more total heat transfer area
3. always uses less live steam and less total area both
4. has no defined economy at all
Show answer
Answer: B — has a higher economy, at the cost of more total heat transfer area
Reusing each effect's vapour as the next effect's heating steam is exactly what lets economy climb above the single-effect ceiling of roughly 1 kg vapour per kg live steam — but each additional effect needs its own heat transfer surface operating across a smaller temperature drop, so total area rises even as live-steam consumption falls.
A heat exchanger transfers 100 kW of duty with an overall heat transfer coefficient U = 500 W/(m2.K) and a log-mean temperature difference of 20 K. What heat transfer area, in m2, is required (Q = UALMTD)?
Numerical answer — type the value.
Show answer
Answer: 10
A = Q/(U*LMTD) = 100,000 W / (500 W/(m2.K) * 20 K) = 100,000 / 10,000 = 10 m2. This is exactly the sizing calculation every double-pipe or shell-and-tube exchanger design boils down to once U and the driving temperature difference are known.
For the same fluid at the same velocity, a large Biot number combined with a small Prandtl number would together suggest
1. the lumped-capacitance model applies and the thermal boundary layer is thicker than the velocity boundary layer
2. internal conduction resistance in the solid cannot be neglected, while the thermal boundary layer in the fluid runs thicker than the velocity boundary layer
3. both quantities describe the same physical ratio and cannot differ from each other
4. radiation must dominate over conduction and convection
Show answer
Answer: B — internal conduction resistance in the solid cannot be neglected, while the thermal boundary layer in the fluid runs thicker than the velocity boundary layer
Biot number (h*Lc/k_solid) and Prandtl number (momentum diffusivity/thermal diffusivity of the fluid) describe two entirely different ratios, in two different media, and can vary independently: a large Bi means the solid's own internal conduction resistance matters and the lumped model fails, while a small Pr (thermal diffusivity exceeding momentum diffusivity, typical of liquid metals) means the thermal boundary layer in the fluid grows thicker than the velocity boundary layer, not thinner.