Fluid Dynamics problem, need help

I have the following thermal/fluids problem for class, and I am stumped. I thought maybe I could use Bernoulli's, but I am not sure if I can assume it's incompressible. If I do use Bernoulli's, I wouldn't be sure how to get the velocity of the water, leaving me with two unknowns still. Trying to use enthalpy doesn't seem to help, as the water is the same temp throughout the process, and if I assume incompressible it would also have the same specific volume, so it seems to go nowhere.

Anyway, here is the problem. Any help would be greatly appreciated:

The entire process is isothermal. The problem statement is repeated here for your convenience. In this new situation, all we know about the state at Point 2 is that P2 = Pa = 100 kPa, and of course, the process is isothermal. MUST USE EES to determine properties. (Hint: visit the EES help on LMS to recall EES definitions for the quality and State).

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a) Neglecting losses, compute the water level h (m) in the figure above for which the water will begin to form vapor cavities (ie. will begin to boil) at the throat of the diffuser (1).

b) A section of pipe of diameter = 8 cm is then added to this device at Point 2. Determine whether the flow is laminar or turbulent and the length it must be in order for the flow to be fully-developed.
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c) If the air surrounding the pipe is just above freezing (1 ï‚°C) and V = 5 m/s, determine the rate of heat transfer from the pipe to the air (kW). Assume constant wall temperature with the outside surface temperature of the pipe equal to the temperature of the water inside the pipe.

a)
1. Water is incompressible.
2, Given that losses are negligible, Bernoulli's theorem applies.
3. Venturi meter calculations are applicable.
4. As you said there are two unknowns a) head causing flow and b) velocity.
5. v(inlet) is given by applying Bernoulli's at that point. (v(inlet)^2/2g) = h m. So, v(inlet) = Sqrt(2gh)
6. v(1) = v(inlet) x (8/5)^2 = 12.8v(inlet) .....by continuity theorem
7. For cavitation to start the pressure at the throat has to be below the saturated vapour pressure of water at 30C P(1) = 4.2 kPa = 0.41 m water
8. P(inlet) = h m
9. Neglecting losses apply Bernoulli's to inlet and the throat:

h + (v(inlet)^2/2g) = 0.41 + (v(1)^2/2g) substitute for v(inlet) from (5) above and for v(1) from (6). You will have a single equation in h. Solve for h.

b) Usually one needs a pipe length of 5 to 10 diameters (in this case 40 to 80cms) for fully developed flow.

c) assuming that the pipe surface is at 30C the temperature differential = 30 - 1 = 29C. If the velocity given as 5m/s is that of the air current you will have to look up the convection heat transfer coefficient h for these conditions.
Q = 29 A h kcal/hr where A = area of pipe
KW = Q/860