What is the critical radius here and what does it mean?

7 mm; insulation always helps since pipe radius is larger

7 mm; insulation increases loss for this pipe

0.7 m; insulation helps only a little

Practice · 8 marks

Heat loss from insulated steam pipe

Short answer

Radial resistances: convection inside 0.0398, steel 0.00042, insulation 1.6989, air 0.1675 K/W per m. Total 1.9066 K/W gives Q/L = (250−25)/1.9066 = 118 W/m. Pipe/insulation interface ~245.3 °C, outer insulation surface 44.8 °C. Critical radius rc = k_ins/ho = 0.07/10 = 7 mm.

A steel steam pipe of inner diameter 80 mm and outer diameter 90 mm (k = 45 W/m K) is covered with 50 mm of insulation (k = 0.07 W/m K). Steam at 250 °C flows inside with h_i = 100 W/m^2 K. The outside air is at 25 °C with h_o = 10 W/m^2 K. Calculate (a) the heat loss per metre length of pipe, (b) the temperature at the pipe/insulation interface and at the outer surface of insulation, and (c) the critical radius of insulation for this pipe.

Data: $r_1 = 0.04$ m, $r_2 = 0.045$ m, $r_3 = 0.045+0.05 = 0.095$ m. For a cylinder, $R_{cond} = \dfrac{\ln(r_{o}/r_{i})}{2\pi kL}$ and $R_{conv} = \dfrac{1}{hA}$ . Take $L = 1$ m.

Resistances (K/W per metre)

Element	Expression	Value
Steam film	$1/(100\cdot 2\pi\cdot0.04)$	0.03979
Steel wall	$\ln(0.045/0.04)/(2\pi\cdot45)$	0.00042
Insulation	$\ln(0.095/0.045)/(2\pi\cdot0.07)$	1.69890
Outer air film	$1/(10\cdot2\pi\cdot0.095)$	0.16753
Total		1.90663

(a) Heat loss

Q/L = \frac{250-25}{1.90663} = 118.0\ \text{W/m}

(b) Temperatures

\begin{aligned} T_{\text{inner steel}} &= 250 - 118.0(0.03979) = 245.3\ ^\circ\text{C}\\ T_{\text{pipe/insulation}} &= 245.3 - 118.0(0.00042) = 245.26\ ^\circ\text{C}\\ T_{\text{outer insulation}} &= 245.26 - 118.0(1.6989) = 44.8\ ^\circ\text{C} \end{aligned}

Check: $44.8 - 118.0(0.16753) = 25.0$ °C.

(c) Critical radius

r_c = \frac{k_{ins}}{h_o} = \frac{0.07}{10} = 0.007\ \text{m} = 7\ \text{mm}

Since the pipe outer radius (45 mm) is already much greater than $r_c$ , adding insulation always reduces heat loss here.

Answer: $Q/L \approx 118\ \text{W/m}$ ; interface 245.3 °C; insulation outer surface 44.8 °C; $r_c = 7$ mm.

✓ Answer checked by Claude Sonnet 5.5

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All Heat Transfer chapter 2 questions →

Heat loss from insulated steam pipe

Resistances (K/W per metre)

(a) Heat loss

(b) Temperatures

(c) Critical radius

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