22-Agric-B8 Food Process Engineering (Part 1) · May 2017
Question 2 of 10: Steam-pipe heat loss and the payback of adding insulation
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
04-Agric-B8, Food Process Engineering (Part 1) — National Exams, May 2017. 3 hours duration, closed book (one aid sheet permitted). Ten questions are printed, grouped into four sections (I–IV); a complete exam paper requires six. All ten are solved below as a complete study set.
Reference texts: Toledo, R.T., Fundamentals of Food Process Engineering (this exam's own cited source for its Stumbo g-table and steam-table appendix); Geankoplis, C.J., Transport Processes and Separation Process Principles (evaporator design and steam economy); Incropera, F.P. & DeWitt, D.P., Fundamentals of Heat and Mass Transfer (unsteady-state Heisler-chart conduction, composite cylindrical walls); Singh, R.P. & Heldman, D.R., Introduction to Food Engineering, and Cleland, A.C., Food Refrigeration Processes (freezing-time prediction, modified Plank equation).
Question 2: Steam-pipe heat loss and the payback of adding insulation (15 marks)
Given. A steel pipe carries saturated steam at 130°C; heat is lost radially through the pipe wall and (in part b) an added insulation layer to 15°C ambient air.
Given data
Quantity
Symbol
Value
Pipe inside diameter
D₁
4.09 cm (r₁=0.02045 m)
Pipe outside diameter
D₂
4.826 cm (r₂=0.02413 m)
Inside (steam) film coeff.
hᵢ
11 400 W/(m²·K)
Outside (air) film coeff.
hₖ
5.7 W/(m²·K)
Steel conductivity
kᶿ
45 W/(m·K)
Steam / air temperature
Tᵢ, Tₖ
130°C, 15°C
Insulation (part b)
k᷀, thickness
0.07 W/(m·K), 5 cm
Find. (a) Heat loss per metre of bare pipe and steam condensation rate; (b) annual energy saved per metre once 5 cm of insulation is added.
Radial resistance network for the bare pipe (r₁→r₂) and, in part (b), the added insulation annulus (r₂→r₃).
Approach. Series resistances per unit pipe length: convection inside → conduction through the steel wall → convection outside (and, in part b, a further insulation-conduction resistance in series before the outside film); Q/L = ΔT/ΣR.
Part (a) — resistance network, bare pipe. Per unit length: $$R_i=\frac{1}{h_i 2\pi r_1},\quad R_{wall}=\frac{\ln(r_2/r_1)}{2\pi k_{steel}},\quad R_o=\frac{1}{h_o 2\pi r_2}$$
Rᵢ = 1/(11400×2π×0.02045) = 6.83×10⁻&sup4; m·K/W. R𝑤all = ln(0.02413/0.02045)/(2π×45) = 5.85×10⁻&sup4;. Rₖ = 1/(5.7×2π×0.02413) = 1.1571 m·K/W. The outside film dominates completely (>99.7% of the total resistance) — a thin steel wall with a strong inside film transfers heat almost freely; the bottleneck is the outside air film.
Heat loss and condensation rate. ΣR = 0.000683+0.000585+1.1571 = 1.1584 m·K/W.
$$\frac{Q}{L}=\frac{T_i-T_o}{\Sigma R}=\frac{130-15}{1.1584}=\boxed{99.3\ \text{W/m}}$$
At 130°C, h᷇ᵤ = 2174.2 kJ/kg (steam tables). Condensation rate per metre:
$$\dot m = \frac{(Q/L)\times 3600}{h_{fg}} = \frac{99.3\times 3600}{2\,174\,200} = \boxed{0.164\ \text{kg/(h}\cdot\text{m)}}$$
Part (b) — add the insulation annulus. Outer insulation radius r₃ = r₂+0.05 = 0.07413 m.
$$R_{ins}=\frac{\ln(r_3/r_2)}{2\pi k_{ins}}=\frac{\ln(0.07413/0.02413)}{2\pi(0.07)}=2.552\ \text{m}\cdot\text{K/W},\qquad R_{o,new}=\frac{1}{h_o 2\pi r_3}=0.377\ \text{m}\cdot\text{K/W}$$
New total: ΣR₂ = 0.000683+0.000585+2.552+0.377 = 2.930 m·K/W.
$$\left(\frac{Q}{L}\right)_{new}=\frac{130-15}{2.930}=\boxed{39.3\ \text{W/m}}$$
Annual energy saved. Δ(Q/L) = 99.3−39.3 = 60.0 W/m. Over one year (8760 h):
$$E_{saved}=60.0\ \text{W/m}\times 8760\ \text{h}=525.8\ \text{kWh per metre of pipe, per year}$$
Even with a modest 5 cm insulation thickness and a fairly conductive foam (k=0.07), the outside film resistance dominates so strongly that adding insulation still cuts total heat loss by 60% — insulation is worthwhile even though the bare-pipe loss was already film-limited.
