SURVIVAL MODE Quiz 3 – 2013.11.29

Question Liquid water at 1 atm is being pumped at a rate of 0.189 m3/s from a large storage tank by a pump with a rating of 2 kW. The water is pumped through a heat exchanger, where it gives up 758 kW of heat and is then delivered to a storage tank at 18.76 m above the first tank. What is the total change in temperature for the water? For water:  = 1 g/cm3, cP = 4.184 J/g∙K. TIME IS UP!!!

Solution   Assumptions: Steady state process No heat loss in the tanks and along the pipes Uniform pipe diameter Constant fluid properties  Q ∆z  WS

Solution

Solution

Outline Mass Balance Energy Balance Momentum Balance

Momentum Balance Change in pressure forces Rate of increase of momentum Rate of change in momentum Force of solid surface on fluid Force of gravity on fluid

Momentum Balance We can rewrite the following terms as: Substituting into the balance equaton:

Momentum Balance For turbulent flows:

Momentum Balance Rewriting:

Momentum Balance Important Notes: All terms are considered vectors, so the direction must be specified (x, y, or z). The force due to gravity only acts along the y-direction. This equation assumes that the flow is turbulent, and the velocity profile is flat.

Exercise A diagram of a liquid-liquid ejector is shown in the figure below. It is desired to analyze the steady-state mixing of two streams, both of the same fluid, by means of overall balances. At plane 1 the two fluids merge. Stream 1a has a velocity v0 and a cross-sectional area (1/3)A1, and Stream 1b has a velocity (1/2)v0 and a cross-sectional area (2/3)A1. Plane 2 is chosen far enough downstream so that the two streams have mixed and the velocity is almost uniform at v2. The flow is turbulent and the velocity profiles at planes 1 and 2 are assumed to be flat. Neglect Fs→f and gravity effects.

Liquid-Liquid Ejector We can rewrite the entire system like this: Assumptions: Steady-state flow A1 = A2 (cross-sectional area) Incompressible fluid Unidirectional flow No gravity effects No Fs→f v0 (1/2)v0 Plane 1 Plane 2

Overall Mass Balance Assumptions: Steady-state flow A1 = A2 (cross-sectional area) Incompressible fluid Unidirectional flow No gravity effects No Fs→f v0 (1/2)v0 Plane 1 Plane 2

Overall Momentum Balance Assumptions: Steady-state flow A1 = A2 (cross-sectional area) Incompressible fluid Unidirectional flow No gravity effects No Fs→f Since the flow is turbulent and unidirectional:

Overall Momentum Balance Assumptions: Steady-state flow A1 = A2 (cross-sectional area) Incompressible fluid Unidirectional flow No gravity effects No Fsf Questions: What conclusion can be made from the above result? If we are to carry out an MEB on the system (ΣF is significant), what result should we expect? What is ΣF is negligible?

Exercise Fluid is flowing at steady state through a reducing pipe bend, as shown in the figure below. Turbulent flow will be assumed with frictional forces negligible. The volumetric flow rate of the liquid and the pressure p2 at point 2 are known, as are the pipe diameters at both ends. Derive the equations to calculate the forces on the bend. Assume that the fluid density is constant.

Exercise Required Quantity: Force of the fluid on the surface Assumptions: Steady-state flow Incompressible fluid Only the x- and the y-direction are involved Significant gravity effects No friction loss

Overall Mass Balance From the mass balance, given the volumetric flowrate and areas of the bend, we can obtain the velocities at the two points.

Mechanical Energy Balance Assumptions: Steady-state flow Incompressible fluid Only the x- and the y-direction are involved Significant gravity effects No friction loss From the energy balance, we now have pressure values.

Overall Momentum Balance Assumptions: Steady-state flow Incompressible fluid Only the x- and the y-direction are involved Significant gravity effects No friction loss Resolving the forces into its x- and y-components:

Overall Momentum Balance Based on the figure, the unit vectors are: Plugging in the unit vectors:

Overall Momentum Balance The force exerted by the fluid on the bend have components: Questions: What would be the magnitude and direction of this force? What will be the force exerted by the bend on the fluid?

Exercise Water at 95°C is flowing at a rate of 2.0 ft3/s through a 60° bend, in which there is a contraction from 4 to 3 inches internal diameter. Compute the force exerted on the bend if the pressure at the downstream end is 1.1 atm. The density and viscosity of water at the conditions of the system are 0.962 g/cm3 and 0.299 cp, respectively.