Chapter 11: Heat Exchangers

Sources: brief Biddle comments in Heat Transfer (36); related U/UA from (07), (29), (30); textbook Chapter 11 (11.1–11.4) as primary support. No worked examples included.

1. Overview

A heat exchanger moves heat from a hot fluid to a cold fluid through a separating wall: hot-fluid convection → wall conduction → cold-fluid convection. This is why Ch. 11 leans on the $U$/$UA$ ideas from Chapters 3 and 8.

Two analysis methods:

1. LMTD — terminal (inlet/outlet) temperatures known or found from energy balances.
2. Effectiveness–NTU — outlet temperatures unknown but size/$UA$ known.

Parallel flow: fluids enter the same end, same direction. Counterflow: fluids enter opposite ends. Counterflow is generally more effective because the fluid-to-fluid $\Delta T$ stays more uniform along the length. Because that $\Delta T$ varies along the exchanger, the driving difference is the log-mean $\Delta T_{\text{lm}}$, not an arithmetic average.

2. Governing Equations

2.1 Energy balances & heat capacity rate

With negligible loss to surroundings, heat lost by the hot fluid = heat gained by the cold fluid:

$$\dot{q}=\dot{m}_h c_{p,h}(T_{h,i}-T_{h,o})=\dot{m}_c c_{p,c}(T_{c,o}-T_{c,i})$$

Heat capacity rate $C=\dot{m}c_p$, $[C]=\text{W/K}$, so $C_h=\dot{m}_h c_{p,h}$, $C_c=\dot{m}_c c_{p,c}$:

$$\dot{q}=C_h(T_{h,i}-T_{h,o})=C_c(T_{c,o}-T_{c,i})$$

The fluid with the smaller $C$ changes temperature more for the same $\dot q$.

2.2 Rate equation & overall coefficient $U$

$$\dot{q}=UA\,\Delta T$$

Series resistances for a tube wall (with fouling):

$$\frac{1}{UA}=\frac{1}{h_iA_i}+R_{f,i}+\frac{\ln(r_o/r_i)}{2\pi kL}+R_{f,o}+\frac{1}{h_oA_o}$$

Neglecting fouling:

$$\frac{1}{UA}=\frac{1}{h_iA_i}+\frac{\ln(r_o/r_i)}{2\pi kL}+\frac{1}{h_oA_o}$$

$U$ depends on the reference area, but $UA$ is unique:

$$\frac{1}{UA}=\frac{1}{U_iA_i}=\frac{1}{U_oA_o},\qquad A_i=\pi D_i L,\quad A_o=\pi D_o L$$

With finned surfaces, use the overall surface (temperature) efficiency $\eta_o$:

$$\frac{1}{UA}=\frac{1}{(\eta_o hA)_h}+\frac{R''_{f,h}}{(\eta_o A)_h}+R_w+\frac{R''_{f,c}}{(\eta_o A)_c}+\frac{1}{(\eta_o hA)_c}$$

$R_w$ = wall conduction resistance; subscripts $h,c$ = hot/cold sides ($\eta_o$ from Ch. 3.6.4).

2.3 Log-mean temperature difference (LMTD)

$$\Delta T_{\text{lm}}=\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$$

If $\Delta T_1=\Delta T_2$ then $\Delta T_{\text{lm}}=\Delta T_1=\Delta T_2$. End differences depend on arrangement:

• Parallel: $\Delta T_1=T_{h,i}-T_{c,i}$, $\Delta T_2=T_{h,o}-T_{c,o}$
• Counterflow: $\Delta T_1=T_{h,i}-T_{c,o}$, $\Delta T_2=T_{h,o}-T_{c,i}$ (the reference $\Delta T_{\text{lm,CF}}$)

2.4 LMTD method & correction factor $F$

$$\dot{q}=UA\,\Delta T_{\text{lm}}$$

For multipass / shell-and-tube / crossflow, correct the counterflow LMTD:

$$\dot{q}=UAF\,\Delta T_{\text{lm,CF}}$$

$F=1$ for ideal double-pipe parallel and counterflow. Otherwise read $F=f(P,R)$ from the textbook charts, with $t$ = tube-side, $T$ = shell-side fluid:

$$R=\frac{T_i-T_o}{t_o-t_i},\qquad P=\frac{t_o-t_i}{T_i-t_i}$$

2.5 Effectiveness–NTU (core)

$$\varepsilon=\frac{\dot{q}}{\dot{q}_{\text{max}}},\qquad \dot{q}_{\text{max}}=C_{\text{min}}(T_{h,i}-T_{c,i}),\qquad C_{\text{min}}=\min(C_h,C_c)$$ $$\dot{q}=\varepsilon C_{\text{min}}(T_{h,i}-T_{c,i})$$ $$\text{NTU}=\frac{UA}{C_{\text{min}}},\qquad C_r=\frac{C_{\text{min}}}{C_{\text{max}}},\qquad C_{\text{max}}=\max(C_h,C_c)$$

For every arrangement $\varepsilon=f(\text{NTU},C_r)$ — specific relations in §3.

3. Heat Exchanger Types & Per-Type Formulas

Quick lookup — full ε-NTU relations follow in §3.1–3.5 (end differences for $\Delta T_{\text{lm}}$ are in §2.3):

Type	Key feature	Rate form	F
Double-pipe parallel flow	fluids enter same end, same direction	$\dot q=UA\,\Delta T_{\text{lm}}$	$F=1$
Double-pipe counterflow	fluids enter opposite ends	$\dot q=UA\,\Delta T_{\text{lm,CF}}$	$F=1$ (reference)
Shell-and-tube	one fluid in tubes, one in shell; baffles induce shell-side crossflow	$\dot q=UAF\,\Delta T_{\text{lm,CF}}$	chart ($P,R$)
Crossflow (single pass)	fluids cross ≈ perpendicular; mixed/unmixed behavior matters	$\dot q=UAF\,\Delta T_{\text{lm,CF}}$	chart ($P,R$)
Compact exchanger	large area density (>700 m²/m³), small passages, often gas-side controlling	$\dot q=UA\,\Delta T_{\text{lm}}$ (via its arrangement)	per arrangement

3.1 Double-pipe parallel flow

$$\varepsilon=\frac{1-e^{-NTU(1+C_r)}}{1+C_r}\qquad(11.28a)$$

3.2 Double-pipe counterflow

$$\varepsilon=\frac{1-e^{-NTU(1-C_r)}}{1-C_r\,e^{-NTU(1-C_r)}}\qquad(C_r<1,\ \ 11.29a)$$ $$\varepsilon=\frac{NTU}{1+NTU}\qquad(C_r=1)$$

3.3 Shell-and-tube

One shell pass (2, 4, … tube passes):

$$\varepsilon_1=2\left\{1+C_r+(1+C_r^2)^{1/2}\,\frac{1+e^{-(NTU)_1(1+C_r^2)^{1/2}}}{1-e^{-(NTU)_1(1+C_r^2)^{1/2}}}\right\}^{-1}\qquad(11.30a)$$

$n$ shell passes (2n, 4n, … tube passes):

$$\varepsilon=\left[\left(\frac{1-\varepsilon_1C_r}{1-\varepsilon_1}\right)^{n}-1\right]\left[\left(\frac{1-\varepsilon_1C_r}{1-\varepsilon_1}\right)^{n}-C_r\right]^{-1}\qquad(11.31a)$$

Correction factor: $R=\dfrac{T_i-T_o}{t_o-t_i}$, $P=\dfrac{t_o-t_i}{T_i-t_i}$; read $F$ from Fig. 11.10 (1 shell, 2/4/… tube passes).

3.4 Crossflow (single pass)

Both fluids unmixed:

$$\varepsilon=1-\exp\left[\frac{1}{C_r}(NTU)^{0.22}\left(e^{-C_r(NTU)^{0.78}}-1\right)\right]\qquad(11.32)$$

$C_{max}$ mixed, $C_{min}$ unmixed:

$$\varepsilon=\frac{1}{C_r}\left(1-e^{-C_r\left[1-e^{-NTU}\right]}\right)\qquad(11.33a)$$

$C_{min}$ mixed, $C_{max}$ unmixed:

$$\varepsilon=1-e^{-C_r^{-1}\left[1-e^{-C_r NTU}\right]}\qquad(11.34a)$$

3.5 Compact exchanger

No distinct relation — analyze with the relation for its actual flow arrangement (usually crossflow).

Any type with $C_r=0$ (one fluid boiling/condensing, $C\to\infty$):

$$\varepsilon=1-\exp(-NTU)\qquad(11.35a)$$

4. Solution Workflows

4.1 LMTD — terminal temperatures known

1. Identify exchanger type / flow arrangement.
2. Compute $C_h$, $C_c$.
3. Energy balances → missing outlet temperature or $\dot q$.
4. Get $\Delta T_1$, $\Delta T_2$ for the arrangement.
5. Compute $\Delta T_{\text{lm}}$.
6. Apply $F$ if shell-and-tube/crossflow.
7. $\dot{q}=UAF\Delta T_{\text{lm}}$ → solve for $\dot q$, $A$, or $U$.

4.2 Effectiveness–NTU — outlets unknown, $UA$ known

1. Compute $C_h$, $C_c$, $C_{\text{min}}$, $C_{\text{max}}$, $C_r$.
2. $\text{NTU}=UA/C_{\text{min}}$.
3. Pick the $\varepsilon$ relation for the type (§3).
4. Compute $\varepsilon$.
5. $\dot{q}=\varepsilon C_{\text{min}}(T_{h,i}-T_{c,i})$.
6. Energy balances → outlet temperatures.

5. Quick Decision Rules

• Terminal temps known / easy from energy balance → LMTD. Outlets unknown with $UA$ known → $\varepsilon$–NTU.
• Sizing $A$ from known terminal temps → LMTD is direct. Predicting outlets of an existing exchanger → $\varepsilon$–NTU is direct.
• Smaller $C$ changes temperature more; $C_{\text{min}}$ sets $\dot q_{\text{max}}=C_{\text{min}}(T_{h,i}-T_{c,i})$.
• Double-pipe parallel/counter → $F=1$. Shell-and-tube or crossflow → use $F$ chart, $\dot{q}=UAF\Delta T_{\text{lm}}$.

6. Key Takeaways

• HEs transfer heat between two fluids; apply energy balances on each.
• $U$ bundles convection + wall conduction (+ fouling, + fin efficiency); $UA$ is the overall conductance and is independent of area basis.
• LMTD handles the along-length variation of $\Delta T$; use when terminal temps are known.
• $\varepsilon$–NTU when outlets are unknown; $C_{\text{min}}$ limits $\dot q_{\text{max}}$.
• Counterflow is generally more effective than parallel flow.
• Correction factor $F$ is needed for most shell-and-tube and crossflow configurations.