Question 5 of 6: Adiabatic Compressible Steam Flow — Normal Delivery and Coil-Shear Relief

Question 5: Adiabatic Compressible Steam Flow — Normal Delivery and Coil-Shear Relief (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Normal steam delivery capacity, intact 73 ft supply line

Given. Saturated steam at $P_1=125$ psig ($139.7$ psia); 0.5-in schedule 80 pipe, internal diameter $D=0.546$ in; total intact flow path = 53 ft header + 20 ft coil $=73$ ft equivalent length; discharge to atmospheric back-pressure $P_{atm}=14.7$ psia; commercial steel pipe (roughness $\varepsilon=0.045$ mm, no roughness value given).

Find. The steady mass flow rate of steam the intact 73 ft supply line can deliver, using adiabatic flow of a compressible gas through a pipe with friction.

Approach. Steam is approximated as an ideal gas ($k=1.3$, $T_{sat}(139.7\ \text{psia})\approx451.4\ \text{K}$, from saturated steam tables) for the compressible-flow calculation. First check whether the flow chokes: the critical pressure ratio for $k=1.3$ is $$\left(\frac{P^*}{P_1}\right)_{crit}=\left(\frac{2}{k+1}\right)^{k/(k-1)}=0.542$$ Since $P_{atm}/P_1=14.7/139.7=0.105\ll0.542$, the flow chokes (reaches sonic velocity) somewhere inside the pipe regardless of how low the downstream pressure actually is, so the maximum mass flux $G$ is governed entirely by the pipe's own friction length $fL/D$ and the header's stagnation state — not by the exact exit back-pressure. Adiabatic Fanno-flow theory (energy balance + momentum balance + ideal-gas EOS, integrated along the duct) gives the sonic-point temperature and the friction-length relation: $$T^*=T_1\frac{2}{k+1},\qquad \frac{fL}{D}=-\frac{k+1}{k}\ln\frac{v^*}{v_1}-\frac{R_sT_1}{G^2}\left(\frac{1}{v^{*2}}-\frac{1}{v_1^{2}}\right),\qquad v^*=\frac{\sqrt{kR_sT^*}}{G}$$ solved simultaneously for $G$ given the pipe's actual $fL/D$ (friction factor $f$ iterated against Reynolds number via the Colebrook/Swamee–Jain correlation, since $f$ depends on the very mass flux being solved for).

1. Sonic-point temperature. Independent of friction, from the energy balance alone: $$T^*=T_1\frac{2}{k+1}=451.4\times\frac{2}{2.3}=\boxed{392.5\ \text{K}}$$
2. Friction factor, iterated. Starting from an assumed $f\approx0.02$ (fully turbulent estimate), solving for $G$, computing $Re=GD/\mu$ ($\mu_{steam}\approx1.3\times10^{-5}\ \text{Pa}\cdot\text{s}$), and re-evaluating $f$ via Swamee–Jain converges after a few iterations to $$Re\approx3.24\times10^5,\qquad f\approx0.0273,\qquad \frac{fL}{D}=0.0273\times\frac{73\times0.3048}{0.546\times0.0254}=43.8$$
3. Solve for the choked mass flux $G$. Substituting $fL/D=43.8$ into the Fanno-flow friction-length relation and solving simultaneously with the sonic-point condition ($v^*=\sqrt{kR_sT^*}/G$) gives $$G=\boxed{304\ \text{kg/(m}^2\text{s)}}$$ with sonic-point pressure $P^*\approx16.5\ \text{psia}$ — confirming $P^*>P_{atm}$, i.e. the flow is indeed choked as assumed.
4. Mass flow rate. Pipe flow area $A=\frac{\pi}{4}D^2=\frac{\pi}{4}(0.546\times0.0254)^2=1.511\times10^{-4}\ \text{m}^2$: $$\dot m=GA=304\times1.511\times10^{-4}=0.0459\ \text{kg/s}=\boxed{364\ \text{lb}_m/\text{hr}}$$

The full 73 ft supply line is friction-limited well before it could deliver anywhere near the flow a short, frictionless opening at 125 psig would suggest — the very high $fL/D\approx44$ for this small-bore (0.546-in) pipe is the dominant term, which is exactly why the shorter path in part (b) delivers a noticeably higher flow for the same upstream conditions.

(b) Maximum mass flow after a coil shear

Given. Same header/steam state as part (a); if the coil shears, the worst case (maximum credible relief flow, the governing case for sizing a relief system) is a shear occurring at the point where the coil meets the reactor wall — i.e. the shortest possible remaining flow path from the header, just the 53 ft header itself, with none of the 20 ft coil length adding further friction.

Find. The maximum mass flow rate of steam discharging from the sheared point, via the same adiabatic-flow-with-friction method as part (a).

1. Friction factor, iterated (53 ft path). Same iteration as part (a) but with $L=53$ ft: $$Re\approx3.76\times10^5,\qquad f\approx0.0272,\qquad \frac{fL}{D}=0.0272\times\frac{53\times0.3048}{0.546\times0.0254}=31.7$$
2. Choked mass flux. Shorter path $\to$ less total friction $\to$ higher achievable $G$ for the same header state: $$G=\boxed{352\ \text{kg/(m}^2\text{s)}}$$ with sonic-point pressure $P^*\approx19.1\ \text{psia}>P_{atm}$ (choked, confirmed).
3. Mass flow rate. $$\dot m=GA=352\times1.511\times10^{-4}=0.0532\ \text{kg/s}=\boxed{422\ \text{lb}_m/\text{hr}}$$

Because friction dominates this small-bore pipe, the maximum credible relief flow (422 lb_m/hr, a shear right at the coil's entry to the vessel) is about 16% higher than the intact system's normal delivery flow (364 lb_m/hr) — a modest, not dramatic, increase, since even the shorter 53 ft path still carries a large friction penalty for a pipe this small.

(c) Relief-system recommendation

Part (c) — With the source's own text for this 2-mark sub-part unrecoverable (see the Source note above), the professional-practice answer consistent with the part (b) result is: specify a spring-loaded safety/relief valve on the reactor, set to open at or below the vessel's maximum allowable working pressure and sized (per API RP 520/526) to pass at least the maximum credible steam flow found in part (b), $\approx422\ \text{lb}_m/\text{hr}$ (with an added margin for accumulation), discharging through a properly sized tailpipe to a safe location; a low-flow or high-temperature interlock on the coil supply line, tripping the header isolation valve, is a complementary detection layer that would limit the duration (not just the peak rate) of any coil-shear release.