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ENGR5002 / Week 4
ENGR271 2026 Q2Past paperCurrent spec2:140 marks48 min

ENGR271 Summer 2026 Q2[VALID]

Water flows from a pressurised tank through a pipe and a converging nozzle before discharging to the atmosphere. The tank free surface (section 1) is at elevation z1=10z_1 = 10 m. The pipe rises to the nozzle, and both section 2 (pipe-nozzle junction) and section 3 (nozzle exit) are at the same elevation z2=z3=20z_2 = z_3 = 20 m (Figure Q2-1 in the paper).

Pipe: length L=100L = 100 m, diameter D=0.10D = 0.10 m, Darcy friction factor f=0.020f = 0.020, area A2=πD2/4A_2 = \pi D^2/4. Nozzle: exit diameter d=0.050d = 0.050 m, head-loss coefficient K=0.80K = 0.80, exit area A3=πd2/4A_3 = \pi d^2/4. Volumetric flow rate Q=0.010 m3/sQ = 0.010\ \mathrm{m^3/s}. Water: ρ=1000 kg/m3\rho = 1000\ \mathrm{kg/m^3}, μ=1.0×10−3\mu = 1.0 \times 10^{-3} Pa s. g=9.81 m/s2g = 9.81\ \mathrm{m/s^2}.

Figure Q2-1: pressurised tank connected to a pipe and a converging nozzle.
Figure Q2-1: pressurised tank connected to a pipe and a converging nozzle.
Formulas you may need
  • Volume flow rate: Q=vAQ = vA; mass flow rate m˙=ρQ\dot m = \rho Q (on the formula sheet)
  • Reynolds number: Re=ρvLμRe = \dfrac{\rho v L}{\mu}; pipe laminar Re<2400Re < 2400, turbulent Re>4000Re > 4000 (on the formula sheet)
  • Entrance length: laminar LE/D=0.065 ReL_E/D = 0.065\,Re; turbulent LE/D=1.359 Re0.25L_E/D = 1.359\,Re^{0.25} (on the formula sheet)
  • Darcy head loss: hL=fLDv22gh_L = f \dfrac{L}{D} \dfrac{v^2}{2g} (on the formula sheet)
  • Minor loss: hL=KLv22gh_L = K_L \dfrac{v^2}{2g} (on the formula sheet; also printed in part (d))
  • Hagen-Poiseuille: Δp=8μLQπR4=32μLvavgD2\Delta p = \dfrac{8 \mu L Q}{\pi R^4} = \dfrac{32 \mu L v_{avg}}{D^2} (laminar only) (on the formula sheet)
  • Bernoulli with head loss: v122g+z1+p1ρg=v222g+z2+p2ρg+hL\dfrac{v_1^2}{2g} + z_1 + \dfrac{p_1}{\rho g} = \dfrac{v_2^2}{2g} + z_2 + \dfrac{p_2}{\rho g} + h_L (on the formula sheet)
  • Momentum balance: (m˙v⃗)in−(m˙v⃗)out+∑F⃗ext=ddt(mv⃗)CV(\dot m \vec v)_{in} - (\dot m \vec v)_{out} + \sum \vec F_{ext} = \dfrac{d}{dt}(m \vec v)_{CV}, with ∑F⃗ext=F⃗g+F⃗p+F⃗μ+F⃗others\sum \vec F_{ext} = \vec F_g + \vec F_p + \vec F_\mu + \vec F_{others} (on the formula sheet)
  1. (a)
    Calculate the average velocity in section 3 (nozzle exit).
    [5]
  2. (b)
    Calculate the flow rate and the fluid velocity in section 2.
    [5]
  3. (c)
    Calculate the Reynolds number in the pipe (excluding the nozzle). Is the flow laminar or turbulent? Using your engineering judgment, explain if considering the flow "fully developed" is a good approximation in this case.
    [5]
  4. (d)
    Calculate the head loss in the nozzle using hL,nozzle=K v322gh_{L,nozzle} = K\,\dfrac{v_3^2}{2g}.
    [5]
  5. (e)
    Calculate the head loss in the pipe. Then discuss and demonstrate whether the Hagen-Poiseuille law can be used to calculate the head loss in this specific case.
    [5]
  6. (f)
    Determine the gauge pressure required at the tank's free surface to sustain the given flow rate. Then comment on which component of the system (the pipe or the nozzle) contributes most to the total energy loss.
    [5]
  7. (g)
    Draw the free-body diagram of the forces around the nozzle (include only the pressure forces at sections 2 and 3 and the force exerted by the nozzle walls on the fluid; neglect gravity and other body forces). Apply the momentum balance equation and determine the force required to restrain the nozzle.
    [10]