Heat Exchanger Calculator: LMTD, NTU, Shell-and-Tube & Plate

Thermal rating and sizing

Heat-capacity rates are $C_h=\dot m_h c_{p,h}$ and $C_c=\dot m_c c_{p,c}$, with $C_r=C_{min}/C_{max}$ and $NTU=UA/C_{min}$. Duty is $Q=\varepsilon C_{min}(T_{h,in}-T_{c,in})$. The counterflow and parallel-flow expressions and the equal-shell series combination follow the standard Incropera formulation, checked against Lienhard & Lienhard, sixth edition, Chapter 3 (attached excerpt pp. 4–8, printed pp. 122–126). Exponential and logarithmic divided differences use expm1 and log1p, with their zero-argument limits evaluated explicitly, to preserve small-NTU and balanced-capacity accuracy. In sizing, the hot-outlet target sets duty, the NTU relation is inverted analytically, and $A=NTU\,C_{min}/U$.

The counterflow reference $\Delta T_{lm}$ uses $T_{h,in}-T_{c,out}$ and $T_{h,out}-T_{c,in}$. Equal end differences use their common value. The computed correction is $F=Q/(UA\Delta T_{lm})$, so $Q=UA F\Delta T_{lm}$. This is the shell correction obtained from the same analytical thermal model, without TEMA or VDI charts. For parallel flow, F is explicitly relative to counterflow LMTD.

Kern shell and turbulent tubes

Kern (1950) geometry and correlations are checked against the open shell-and-tube course chapter, attached pp. 8–14: $A_s=D_sB(p_t-d_o)/p_t$; $D_e$ is four times unit-cell free area divided by tube wetted perimeter. $Nu_s=0.36Re_s^{0.55}Pr_s^{1/3}(\mu_b/\mu_w)^{0.14}$. The smooth-tube Petukhov–Kirillov expression uses $f_D=4(1.58\ln Re_t-3.28)^{-2}$, with the 1.07 denominator form shown on attached p. 5.

Tube outside-area resistance is $1/U_o=1/h_s+R_{f,h}+(d_o/d_i)(1/h_t+R_{f,c})+d_o\ln(d_o/d_i)/(2k_w)$. Kern shell friction is $f_s=\exp(0.576-0.19\ln Re_s)$; the number of crossings is approximated by active length / baffle spacing. Tube losses include four velocity heads per pass. Series shell losses add.

Martin plates

Martin's 1996 framework is implemented in the explicit form published by the author in 1999, attached p. 6, Appendix A–H: Fanning friction factor, $d_h=2b/\Phi$, and $Nu=0.205Pr^{1/3}(\mu_b/\mu_w)^{1/6}[f_F Re^2\sin(2\beta)]^{0.374}$. Plate channel loss is $4f_F(L/d_h)G^2/(2\rho)$. For equal channel counts n, there are $2n+1$ plates and $2n-1$ active separating plates; area is $(2n-1)\Phi WL$. Plate resistance adds both films, wall thickness / conductivity, and both fouling resistances.

Axial temperatures

Profiles are derived here from constant-property differential energy balances. A shell exchanges half its conductance with each tube pass: $C_h T_h'=-UA(T_h-(T_1+T_2)/2)$ and $C_c T_1'=(UA/2)(T_h-T_1)$, $C_c T_2'=-(UA/2)(T_h-T_2)$, with normalized local axial position. The tube passes join at the turnaround. Equal shells are connected countercurrently using the same per-shell effectiveness. More than two tube passes are shown as an equivalent two-pass thermal model. Plate profiles integrate the corresponding coflow or counterflow two-stream balances.

Profile temperature changes use the module heat duty directly and scaled exponential differences. This preserves the cold inlet, outlet and turnaround boundaries as conductance approaches zero. All arrangements recover the published small-NTU limit $\varepsilon\approx NTU$ (Lienhard, Chapter 3).

• Steady, adiabatic, single-phase sensible heat exchange; constant mean properties, U and flow distribution.
• Hot fluid occupies the shell and cold fluid occupies the tubes. Each shell has the same area and geometry, with full stream flow through every unit. Even tube passes use the ideal one-shell thermal relation.
• Kern geometry assumes a 25% baffle cut and ordinary clearances. Bundle free-area checks are necessary geometry checks, not a complete tube-sheet layout.
• Plates have equal channel counts, one pass on each side and uniform flow. End plates do not transfer heat to the surroundings. Chevron angle is measured from the main flow direction.
• Geometry sizing changes active length at fixed tube/channel counts and cross sections. Specified-U mode uses U exactly as supplied, without adding the separate geometry-mode fouling inputs.
• Mean properties are entered by the user; the default water-like properties are illustrative constants. Wall viscosity corrections are entered ratios, not an automatic wall-temperature iteration.
• Wort preset: hot flow 0.05 kg/s, heat capacity 4000 J/(kg·K), inlet 95 °C; water 0.10 kg/s, 4180 J/(kg·K), inlet 15 °C; assumed U 800 W/m²·K and area 0.60 m². Recovery preset: two 0.50 kg/s water streams, 70/10 °C inlets, assumed U 1000 W/m²·K and area 1.50 m². These are author-defined scenarios, not product data.
• Oil preset reproduces the attached Bannerman example using its rounded 2.04 m² area and specified U. Graph-read answers are compared with explicit rounding tolerances.
• Pressure and active-length result cards show a dash in specified-U mode because no operating passage geometry has been rated. Numeric geometry results follow the metric/US display switch.

Kern shell heat transfer is restricted here to $2000\le Re_s\le10^6$. The tube correlation is used for $10^4\le Re_t\le5\times10^6$, $0.5\le Pr_t\le2000$ and active length / inside diameter at least 10. Baffle spacing is limited to 0.2–1 shell diameters, with at least two bundle crossings. Use specified U for validated cases outside these correlation ranges.

Martin plate estimates are restricted here to $200\le Re\le10000$ on both sides and chevron angles 10–80°. The original friction model changes branch at Re = 2000. The author reports approximately ±20% heat-transfer accuracy and much broader friction uncertainty (1999 paper, p. 6); this is preliminary rating, not a vendor guarantee.

Shell F below 0.75 triggers a design warning. Very large NTU (25 or more) is rejected to avoid numerical loss of resolution near pinches. No boiling, condensation, freezing, non-Newtonian fluids, temperature-dependent property iteration, batch cooldown, mechanical design, vibration assessment or detailed leakage/maldistribution analysis is included.

Kern pressure loss omits nozzles and external piping. Plate pressure loss includes channel friction only, excluding ports, manifolds, elevation and external piping. The equivalent tube-pass profiles omit baffle-scale mixing and pass-specific local details.

For gases in the plate model, set both bulk/wall viscosity ratios to 1, as specified by Martin.

Thermal sizing gives no pressure rating and no sanitation: use equipment rated for the operating pressure and temperature, and sanitize food-contact passages before chilling wort.