📚 Kundu Nomenclature — Full Reference Click to open

📚 Kundu Nomenclature — Full Reference

Symbols follow Kundu, Cohen & Dowling (6e), with common additions from Batchelor and Panton. Greek names shown in gray chips for clarity.

📑 Concept Index (quick clusters)

Symbol	Meaning
u, v, w	Velocity components (x, y, z). See U, V, W.
u	Velocity vector (u, v, w). See U.
U, U_e, U_CL, U_N	Mean/edge/centerline/far-upstream speeds. See U.
x, y, z; x; x_i	Cartesian coordinates; position vector; components. See X, Y, Z.
r	Radial coordinate/distance. See R.
Γ (Gamma), ω (omega)	Circulation; vorticity vector. See G, O.

Symbol	Meaning
p, p₀, p_atm, P	Static / total / atmospheric / mean pressure. See P.
τ (tau), τ_ij, σ_ij (sigma)	Shear stress; stress tensor; Cauchy stress. See T, S.
f, F	Surface traction per area; force vector. See F.
D, D, L	Drag; Lift. See D, L.

Symbol	Meaning
T, T_w, T₀	Temperature; wall / stagnation temperature. See T.
ρ (rho)	Density. See R.
μ (mu), μ_b (mu)	Dynamic viscosity; bulk viscosity. See M.
ν (nu)	Kinematic viscosity. See V.
k	Thermal conductivity (or wavenumber). See K.
c_p, c_v	Specific heats. See C.
q, Q	Speed magnitude / dynamic pressure / heat flux / volumetric flow (context). See Q.

Symbol	Meaning
Re, Pr, Sc, Sh, Nu	Reynolds, Prandtl, Schmidt, Sherwood, Nusselt numbers. See A–Z.
Fr, Ro, Ri, R_f, Ra	Froude, Rossby, Richardson, Flux Richardson, Rayleigh. See A–Z.
Ma (M), Kn, We, Bo, St, Ta	Mach, Knudsen, Weber, Bond, Strouhal, Taylor. See A–Z.

Symbol	Meaning
ε (epsilon), ε (epsilon)	Dissipation rate of TKE (instantaneous / average).
e, e	Turbulent kinetic energy (per mass); average fluctuation energy.
u^*, κ (kappa)	Friction velocity; von Kármán constant.
δ (delta), δ^*, θ (theta)	BL thickness; displacement thickness; momentum thickness.

Symbol	Meaning
D/Dt	Material derivative.
Δ (Delta)	Dilatation symbol (∇·u).
∇·, ∇×, ∇²	Divergence, curl, Laplacian.

Notation (overbars, primes, subscripts, etc.)

Mark	Meaning
f	Background / average value; also Darcy friction factor (context).
f̃	Full-field value.
f′	Derivative of f or perturbation from reference state.
f^*	Complex conjugate or sonic-condition value.
f⁺	Law-of-the-wall value.

A

Symbol	Meaning
α (alpha)	Contact angle; thermal expansion coefficient; angle of rotation/attack; iteration index.
a	Area (triangular); cylinder/sphere radius; amplitude.
a	Generic vector; Lagrangian acceleration.
A	Generic second-order tensor.
A, 𝔄	Constant; area/surface; wing planform area.
A^*	Control surface; sonic throat area.

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B

Symbol	Meaning
β (beta)	Angle of rotation; constituent-density coefficient; convergence-acceleration parameter; variation of Coriolis frequency with latitude; camber parameter.
b	Generic vector; control-surface velocity (in control-volume sketches).
B	Constant; Bernoulli function; log-law intercept parameter.
B, B_ij	Generic second-order (or higher) tensor.
Bo	Bond number. (Bo, not B_o)

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C

Symbol	Meaning
c	Speed of sound; phase speed; chord length (airfoils).
c	Phase-velocity vector.
c_g, c_g	Group-velocity magnitude / vector.
χ (chi)	Scalar stream function (context dependent in figures).
°C	Degrees Celsius.
C	Generic constant; hypotenuse length; closed contour.
Ca	Capillary number.
C_f	Skin-friction coefficient.
C_p	Pressure coefficient.
c_p, c_v	Specific heat at constant pressure / constant volume.
C_D, C_L	Drag & lift coefficients.
C_ij	Direction-cosine matrix between original and rotated axes.
𝒞^±	Characteristic curves along which invariants I^± are constant.

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D

Symbol	Meaning
d	Diameter; distance; fluid-layer depth.
d	Dipole-strength vector; displacement vector.
δ (delta)	Dirac delta; similarity-variable length; boundary-layer thickness; generic length increment; flow-deflection angle.
δ (delta)	Average boundary-layer thickness.
δ^*	Boundary-layer displacement thickness.
δ_ij	Kronecker delta.
δ₉₉	99% boundary-layer thickness.
Δ (Delta)	Dilatation (∇·u) — Batchelor notation.
D	Distance; drag force; diffusion coefficient.
D	Drag force vector.
D_i	Lift-induced drag.
D/Dt	Material derivative.
D_T	Turbulent diffusivity of particles.
𝒟	Generalized field derivative (abstract notation).

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E

Symbol	Meaning
ε (epsilon)	Roughness height; kinetic-energy dissipation rate; a small distance; fineness ratio h/L; downwash angle (context).
ε (epsilon)	Average dissipation rate of turbulent kinetic energy.
ε_T (epsilon)	Average dissipation rate of the variance of temperature fluctuations.
ε_ijk (epsilon)	Alternating (Levi-Civita) tensor.
e	Internal energy per unit mass.
e_i	Unit vector in the i direction.
e	Average kinetic energy of turbulent fluctuations.
E_c	Eckert number.
E_k, E_p	Kinetic / potential energy per unit horizontal area.
E	Numerical error; average energy per unit horizontal area; Ekman number; kinetic energy of the mean flow (context-dependent).
E_F	Time-average energy flux per unit length of wave crest.

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F

Symbol	Meaning
f	Generic function; Maxwell distribution function; Helmholtz free energy per unit mass; longitudinal correlation coefficient; Coriolis frequency; dimensionless friction parameter (context-dependent).
f (Darcy)	Darcy friction factor.
f_i	Unsteady body-force distribution.
φ (phi)	Velocity potential; also an angle (context).
f	Surface (traction) force vector per unit area.
F	Force magnitude; generic flux/field; profile function.
F_f	Perimeter friction force.
F	Force vector; average wave-energy flux vector.
𝓕	Body-force potential; undetermined spectrum function (context).
F_D	Drag force (instantaneous/average, context).
F_L	Lift force.
Fr	Froude number.

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G

Symbol	Meaning
γ (gamma)	Ratio of specific heats; velocity gradient; vortex-sheet strength; generic dependent variable.
γ̇ (gamma)	Shear rate.
g	Body force per unit mass (e.g., gravity).
g	Acceleration of gravity; undetermined function; transverse correlation coefficient (context).
g′	Reduced gravity.
Γ (Gamma)	Circulation.
G_a	Adiabatic vertical temperature gradient (lapse rate).
Γ_a (Gamma)	Circulation due to absolute vorticity.
G	Gravitational constant; profile function (context).
G_n	Fourier-series coefficient.
G	Center of mass / center of vorticity (context).

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H

Symbol	Meaning
h	Enthalpy per unit mass; height; gap height; viscous-layer thickness.
ℏ	Planck’s constant (reduced).
η (eta)	Free-surface shape; waveform; similarity variable; Kolmogorov microscale (context).
η_T (eta)	Batchelor microscale.
H	Atmospheric scale height; water depth; step function; shape factor; profile function (context).

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I

Symbol	Meaning
I	Moment of inertia (context); integral/indicator (context); identity in some texts (else use δ_ij).
I	Identity tensor (sometimes 𝕀); often implied by δ_ij.
I^±	Characteristic invariants along 𝒞^± (1-D compressible waves).

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J

Symbol	Meaning
J	Jacobian determinant (coordinate transforms); jet momentum flux (context).
J_m	Diffusive mass-flux vector (species transport).
J_n	Bessel function (order n)—pipe/duct eigenproblems.

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K

Symbol	Meaning
k	Thermal conductivity; wavenumber magnitude (context).
κ (kappa)	von Kármán constant (~0.41) in wall turbulence.
K	Bulk modulus; kinetic energy (context).
Kn	Knudsen number.

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L

Symbol	Meaning
ℓ	Characteristic length / mean free path / mixing length (context).
L	Length; lift; Rossby radius; aspect ratio (context).
L_M	Monin–Obukhov length (stratified BLs).

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M

Symbol	Meaning
m	Mass; slope/index (context).
μ (mu)	Dynamic viscosity.
μ_b (mu)	Bulk (second) viscosity.
M	Mach number; mass (context).
M_w	Molecular weight.

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N

Symbol	Meaning
n	Number density; unit normal (vector) (context).
N	Brunt–Väisälä (buoyancy) frequency.
Nu	Nusselt number.
ν (nu)	Kinematic viscosity (also listed under V).

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O

Symbol	Meaning
O	Origin (in sketches/derivations).
Ω (Omega)	Planetary rotation rate / angular velocity of a rotating frame.
ω (omega)	Vorticity vector; angular frequency (context).

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P

Symbol	Meaning
p	Static pressure.
p₀	Total (stagnation) pressure; or reference pressure at z=0 (context).
p_atm	Atmospheric pressure.
P	Power; average/mean pressure (context).
Pr	Prandtl number.
ψ (psi)	Stream function (2-D incompressible; vector Ψ in some contexts).
Π (Pi)	Π-groups (dimensionless products) in Buckingham-Π analysis.

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Q

Symbol	Meaning
Q	Volumetric flow rate; total heat per unit mass (context).
q	Dynamic pressure q = ½ ρ U² (aero); also generic heat flux (context).
q (Batchelor)	Speed magnitude \|u\|.
q_i	Heat-flux vector components.
q_s	2-D point source strength.

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R

Symbol	Meaning
r	Radial coordinate / distance from an origin or axis.
ρ (rho)	Density.
R	Gas constant (specific); radius (context).
R_u	Universal gas constant.
R_i	Radius of curvature (streamline geometry).
Re	Reynolds number.
Ra	Rayleigh number.
Ri	Richardson number.
R_f	Flux Richardson number.
Ro	Rossby number.

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S

Symbol	Meaning
s	Entropy; arc length; salinity; wingspan (aero) — context dependent.
s (vector)	Relative position / displacement vector.
S	Area; salinity; scattered-light intensity (context).
σ (sigma)	Surface tension; normal stress (context).
σ_ij (sigma)	Cauchy stress tensor components (Batchelor).
e_ij	Rate-of-strain tensor (Batchelor notation).
S_ij	Strain-rate (rate-of-strain) tensor.
Sc	Schmidt number.
Sh	Sherwood number.
St	Strouhal number.

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T

Symbol	Meaning
t	Time.
T	Temperature.
θ (theta)	Potential temperature; generic angle (context).
τ (tau)	Shear stress; relaxation time (context).
τ_ij	Stress tensor components.
t_i	Traction component: t_i = σ_ij n_j.
T_w, T₀	Wall temperature; stagnation (total) temperature.
Ta	Taylor number.

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U

Symbol	Meaning
u	Velocity component along x (streamwise).
u	Velocity vector (u, v, w).
u^*	Friction velocity √(τ_w/ρ).
U	Mean / outer / free-stream speed (context).
U_e, U_CL, U_N	Edge velocity (BL); centerline velocity (pipe/jet); far-upstream velocity.

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V

Symbol	Meaning
v	Velocity component along y (often wall-normal in BL); molecular speed (kinetic theory).
v	Molecular velocity vector (kinetic theory); generic vector.
V	Volume; material volume; average cross-stream velocity; average velocity magnitude; complex velocity (context).
V^*	Control volume.
ν (nu)	Kinematic viscosity (also listed under N).
v	Mean/averaged velocity (context).

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W

Symbol	Meaning
w	Vertical velocity component; downwash velocity (aero); complex potential W (context).
W	Thermodynamic work per unit mass; wake function (context).
Ẇ	Rate of energy input from mean flow (turbulence budgets).
We	Weber number.

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X

Symbol	Meaning
x	First Cartesian coordinate (streamwise).
x	Position vector.
x_i	Components of position vector.
ξ (xi)	Similarity variable / coordinate in some transforms (context).

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Y

Symbol	Meaning
y	Second Cartesian coordinate (wall-normal in BL).
Y	Mass fraction.
Y_CL	Centerline mass fraction (jet mixing problems).
Υ (Upsilon)	Used rarely; problem-dependent constant (context).

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Z

Symbol	Meaning
z	Third Cartesian coordinate; complex variable (potential flow).
ζ (zeta)	Interface displacement (free surface); relative vorticity (geophysical context).
Z	Geopotential height / altitude of a pressure surface (synoptic examples).

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Problem 4.56 — Beginner-Friendly Step-by-Step (Print)

0) The Big Picture

2-D steady conduction in a wall with uniform volumetric generation, \( \dot q = 1\times10^{6}\ \mathrm{W/m^3} \). Grid \( \Delta x=\Delta y=\Delta=0.025\ \mathrm{m} \). Thermal conductivity \( k=10\ \mathrm{W/(m\cdot K)} \).

Given nodal temperatures (°C)

Node	Temp
\(T_2\)	95.47
\(T_3\)	117.3
\(T_5\)	79.79
\(T_6\)	77.29
\(T_8\)	87.28
\(T_{10}\)	77.65

Convection boundaries

Inner (Surface B): \(h_i=500,\ T_{\infty,i}=50^{\circ}\mathrm{C}\).
Outer (Surface A): \(h_o=250,\ T_{\infty,o}=25^{\circ}\mathrm{C}\).

Unknowns

\(T_1, T_4, T_7, T_9\), then \(q'_A\), \(q'_B\), and overall balance.

Handy constants

\( \dfrac{h_i \Delta}{k}=1.25 \)
\( \dfrac{h_o \Delta}{k}=0.625 \)
\( \dfrac{\dot q\,\Delta^2}{4k}=15.625 \)
\( \dfrac{\dot q\,\Delta^2}{2k}=31.25 \)

Legend for sketches: blue = conduction, green dashed = convection, ⚡ = generation.

Variable Definitions

\(T_i\) Temperature at node i (°C)
\(k\) Thermal conductivity (W/m·K)
\(\Delta\) Grid spacing (m)
\(\dot q\) Heat generation rate (W/m³)
\(h_i\) Inner convection coeff (W/m²·K)
\(h_o\) Outer convection coeff (W/m²·K)
\(T_{\infty,i}\) Inner fluid temp (°C)
\(T_{\infty,o}\) Outer fluid temp (°C)
\(A\) Control-volume face area (m²)
\(V\) Control-volume size (m³/unit depth)
\(q'_A\) Heat flux leaving surface A (W/m)
\(q'_B\) Heat flux entering surface B (W/m)

T₇

Quick legend: what all the T’s mean

Symbol	Meaning / location
T₁	Top-left corner of solid (top & left insulated)
T₂	Inner surface B, north of the notch (above T₄)
T₃	Interior west of T₄ (also above T₇)
T₄	Re-entrant corner (north & east faces convect to inner fluid)
T₅	Interior east of T₄ (above T₉)
T₆	Inner-surface node on east side (given)
T₇	Outer surface A corner at bottom-left (south face convects)
T₈	Interior just east of T₇
T₉	Bottom edge (between T₈ and T₁₀), south face convects
T₁₀	Bottom-edge corner at right end of the bottom strip

T∞,i and T∞,o are the inner/outer fluid temps; h_i, h_o their convection coefficients.

1) Universal Recipe (works at any node)

\[ \sum_{\text{faces}} k\frac{A}{L_n}(T_{N}-T_P) \;+\; \sum_{\text{conv}} hA\,(T_\infty - T_P) \;+\; \dot q\,V \;=\; 0 \]

Interior CV: \(V=\Delta^2\); edge: \(V=\tfrac12\Delta^2\); corner: \(V=\tfrac14\Delta^2\); re-entrant (node 4): \(V=\tfrac34\Delta^2\).
Half faces use \(A=\tfrac{\Delta}{2}\); center-to-center gaps are \(\Delta\) (or \(\tfrac{\Delta}{2}\) to a boundary).

2a) Node 1 — Corner (insulated top & left)

East & south conduct to \(T_2\), \(T_3\); west & top insulated (no flux); generation adds \(+\dot q V\).

\[ 0 = k\frac{(\Delta/2)(T_2-T_1)}{\Delta} + k\frac{(\Delta/2)(T_3-T_1)}{\Delta} + \dot q\Big(\tfrac14\Delta^2\Big) \]

\[ \Rightarrow\; T_1 = \frac{T_2+T_3}{2} + \frac{\dot q\,\Delta^2}{4k} \approx \boxed{122.0^\circ\mathrm{C}} \]

2b) Node 4 — Re-entrant corner touching inner fluid (north & east)

\[ T_4=\frac{T_2+2T_3+T_5+2T_8+2\left(\tfrac{h_i\Delta}{k}\right)T_{\infty,i} +\tfrac{3\dot q\,\Delta^2}{2k}} {6+2\left(\tfrac{h_i\Delta}{k}\right)} \approx \boxed{94.50^\circ\mathrm{C}} \]

2c) Node 7 — Outer surface A (corner)

\[ T_7=\frac{T_3+T_8+\left(\tfrac{h_o\Delta}{k}\right)T_{\infty,o} +\tfrac{\dot q\,\Delta^2}{2k}} {2+\left(\tfrac{h_o\Delta}{k}\right)} \approx \boxed{95.80^\circ\mathrm{C}} \]

2d) Node 9 — Outer bottom edge (not a corner)

\[ T_9=\frac{T_5+\tfrac12T_8+\tfrac12T_{10} +\left(\tfrac{h_o\Delta}{k}\right)T_{\infty,o} +\tfrac{\dot q\,\Delta^2}{2k}} {2+\left(\tfrac{h_o\Delta}{k}\right)} \approx \boxed{79.67^\circ\mathrm{C}} \]

3) Surface Heat Rates (per unit length)

Outer surface \(A\) (bottom; nodes 7–10)

\[ \begin{aligned} q'_A &= h_o\!\left[\tfrac{\Delta}{2}(T_7 - T_{\infty,o}) + \Delta(T_8 - T_{\infty,o}) + \Delta(T_9 - T_{\infty,o}) + \tfrac{\Delta}{2}(T_{10} - T_{\infty,o})\right] \\ &\approx \boxed{1117\ \mathrm{W/m}} \end{aligned} \]

Inner surface \(B\) (nodes 2, 4, 5, 6)

\[ \begin{aligned} q'_B &= h_i\!\left[\tfrac{\Delta}{2}(T_{\infty,i}-T_2) + \Delta(T_{\infty,i}-T_5) + \tfrac{\Delta}{2}(T_{\infty,i}-T_6) + \Delta(T_{\infty,i}-T_4)\right] \\ &\approx \boxed{-1383\ \mathrm{W/m}} \end{aligned} \]

4) Energy-Balance Check

\[ \dot E'_{\text{in}}-\dot E'_{\text{out}}+\dot E'_{\text{gen}}=0 \quad\Rightarrow\quad -q'_A+q'_B+\dot q\,V' = 0 \]

\[ -1117 - 1383 + 2500 = 0 \quad \Longrightarrow \quad \text{balanced}\ \checkmark \]

Consistent with \(\dot q=10^6\ \mathrm{W/m^3}\) and \(V' = 2.5\times10^{-3}\ \mathrm{m^2}\) for the L-section.

Click “Print PDF” above. The page will typeset math, scroll to the top, then open your print dialog.

Thermal Interview Crash Sheet

Focus: first principles + hand calculations. Two high-yield scenarios: (1) Lumped-capacitance and (2) Transient slab (“thermal block”). Includes Biot & Fourier checks, quick formulas, and interview checklists.

1) Lumped-Capacitance Heating/Cooling

Use when internal temperature gradients are negligible (body ~ isothermal).

Biot Number Check

\[ Bi \equiv \frac{h L_c}{k} < 0.1 \]

Characteristic length (body dependent), e.g. slab \(L_c = \dfrac{V}{A} = \dfrac{t}{2}\), sphere \(L_c=\dfrac{R}{3}\).

Governing Solution (Newton’s Law of Cooling)

\[ T(t) = T_\infty + \bigl(T_i - T_\infty\bigr)\,e^{-t/\tau}, \qquad \tau = \frac{\rho c_p V}{hA} = \frac{\rho c_p L_c}{h} \]

Interview Checklist

State assumption: “Check \(Bi<0.1\) for lumped validity.”
Compute \(Bi\). If valid → use exponential form; compute time constant \( \tau \).
Report \(T(t)\) or dimensionless \( \theta(t) = \dfrac{T-T_\infty}{T_i-T_\infty} = e^{-t/\tau} \).
Limits: Large Bi → use transient slab (below).

Quick Numeric Example

Given Al sphere: \(k=205\,\mathrm{W/(m\cdot K)}\), \(\rho c_p=2.43\times10^6\,\mathrm{J/(m^3\cdot K)}\), \(h=20\,\mathrm{W/(m^2\cdot K)}\), \(R=25\,\mathrm{mm}\).

\[ L_c=\frac{R}{3}=0.00833\,\text{m},\quad Bi=\frac{hL_c}{k}\approx 0.0008 \ (\text{lumped valid}) \] \[ \tau=\frac{\rho c_p L_c}{h}\approx \frac{2.43\times10^6(0.00833)}{20}\approx 1.01\times10^3\text{ s} \]

Use \( T(t)=T_\infty+(T_i-T_\infty)e^{-t/\tau} \) as needed.

2) Transient Slab (“Thermal Block”) with Convection

Use when internal gradients matter (larger bodies or higher \(h\)). Treat as a slab of half-thickness \(L\), with convection at surfaces.

Decision Flow

Biot: \(Bi = \dfrac{hL}{k}\). If \(Bi<0.1\) → lumped (section 1).
Otherwise use 1-term series or Heisler charts (transient slab).
Fourier: \(Fo=\dfrac{\alpha t}{L^2}\) with \( \alpha=\dfrac{k}{\rho c_p} \).

1-Term Slab Solution (common interview acceptance)

Find \( \lambda_1 \) from the transcendental: \( \lambda_1 \tan\lambda_1 = Bi \).

\[ \theta(x,t) \approx A_1 \cos\!\left(\frac{\lambda_1 x}{L}\right)\, e^{-\lambda_1^2 Fo},\quad A_1=\frac{4\sin\lambda_1}{2\lambda_1+\sin(2\lambda_1)} \]

Centerline \(x=0\): \( \theta(0,t)=A_1 e^{-\lambda_1^2 Fo} \).
Surface \(x=L\): \( \theta(L,t)=A_1 \cos\lambda_1 \, e^{-\lambda_1^2 Fo} \).
\( \theta \equiv \dfrac{T-T_\infty}{T_i-T_\infty} \).

Heat to/from the Fluid (surface flux)

\[ q''(t) = h\,[T_s(t)-T_\infty] = -k\left.\frac{\partial T}{\partial x}\right|_{x=L} = k\,\frac{\lambda_1}{L}\,A_1 \sin\lambda_1 \, e^{-\lambda_1^2 Fo}\,(T_i-T_\infty) \]

Use this when asked for “heat rate to the fluid” at a face.

Interview Micro-Checklist (slab)

Define \(L\) (half-thickness), compute \(Bi\) and \(Fo\).
State the governing form (1-term or charts), solve \( \lambda_1 \tan\lambda_1=Bi \) → get \(A_1\).
Report \(T\) or \( \theta \) at center/surface; give \( q''(t) \) if asked about heat to the fluid.

Quick Formulas

Biot & Fourier

\[ Bi=\frac{hL_c}{k},\qquad Fo=\frac{\alpha t}{L^2},\qquad \alpha=\frac{k}{\rho c_p} \]

Lumped Time Constant

\[ \tau=\frac{\rho c_p V}{hA}=\frac{\rho c_p L_c}{h},\qquad \theta(t)=e^{-t/\tau} \]\

Energy Balance (fast check)

\[ \dot{E}_{in}-\dot{E}_{out}+\dot{E}_{gen}=0 \]

At steady state: net in – out + generation = 0. Use this as a 10-second sanity check on any fin/box/slab total.