Heat exchanger types and LMTD derivation

(a) Classification

By flow arrangement

• Parallel flow: both fluids enter at the same end and flow in the same direction. The temperature difference is largest at inlet; the cold outlet can never exceed the hot outlet. It gives a fairly uniform wall temperature, so it is used in simple heaters and coolers where the wall temperature must be limited.
• Counterflow: fluids flow in opposite directions. Gives the highest mean temperature difference and effectiveness (oil coolers, regenerative systems).
• Cross flow: fluids flow at right angles (car radiator, air coolers), mixed or unmixed.

By construction

• Double pipe (pipe-in-pipe): simple, for small duties (small oil and chemical coolers).
• Shell and tube: a bundle of tubes in a shell with baffles; most common in power and process plants (condensers, boilers).
• Compact (plate-fin): large area per volume, for gases (aircraft, automobiles).
• Plate type: corrugated plates, easy to clean (food, dairy).
• Direct contact (cooling tower) and regenerators (stored heat) are other categories.

(b) LMTD for counterflow

Assumptions: steady state; constant $U$, $c_{ph}$ and $c_{pc}$; no heat loss to surroundings; no phase change; negligible axial conduction.

 Th1 -->--------------------->-- Th2
 Tc2 --<---------------------<-- Tc1
      dA element at distance x

For an element of area $dA$: with $\Delta T = T_h - T_c$:

$$dQ = -\dot m_hc_h\,dT_h = -\dot m_cc_c\,dT_c = U\,\Delta T\,dA$$

Taking $x$ along the hot-fluid direction, $T_h$ and $T_c$ both fall as $x$ increases. Writing $C_h = \dot m_hc_h$ and $C_c = \dot m_cc_c$:

$$d(\Delta T) = dT_h - dT_c = -dQ\left(\frac{1}{C_h}-\frac{1}{C_c}\right)$$

Substituting $dQ = U\Delta T\,dA$:

$$\frac{d(\Delta T)}{\Delta T} = -U\left(\frac{1}{C_h}-\frac{1}{C_c}\right)dA$$

Integrate over the full area, from end 1 ($\Delta T_1 = T_{h1}-T_{c2}$) to end 2 ($\Delta T_2 = T_{h2}-T_{c1}$):

$$\ln\frac{\Delta T_2}{\Delta T_1} = -UA\left(\frac{1}{C_h}-\frac{1}{C_c}\right)$$

The total heat is $Q = C_h(T_{h1}-T_{h2}) = C_c(T_{c2}-T_{c1})$, so $\dfrac{1}{C_h}-\dfrac{1}{C_c} = \dfrac{(T_{h1}-T_{h2})-(T_{c2}-T_{c1})}{Q} = \dfrac{\Delta T_1-\Delta T_2}{Q}$. Therefore

$$\ln\frac{\Delta T_2}{\Delta T_1} = -\frac{UA(\Delta T_1-\Delta T_2)}{Q} \;\Rightarrow\; Q = UA\,\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)} = UA\,\Delta T_{lm}$$ $$\boxed{\Delta T_{lm}=\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)}}$$

The same expression holds for parallel flow, with $\Delta T_1$, $\Delta T_2$ taken at the inlet and outlet ends. If $\Delta T_1 = \Delta T_2$ then $\Delta T_{lm} = \Delta T_1$. For cross-flow and multi-pass exchangers, $Q = UAF\Delta T_{lm,cf}$ with a correction factor $F<1$.