2A4A Applied Fluid Mechanics I 2024 PDF

2A4A 2024 Part (A) Applied Fluid Mechanics: Potential flow Part (B): Heat Exchangers. Notes and corrections to [email protected] Note to students. This tutorial sheet covers two distinct areas of the course: potential flow theor...

2A4A 2024 Part (A) Applied Fluid Mechanics: Potential flow Part (B): Heat Exchangers. Notes and corrections to [email protected] Note to students. This tutorial sheet covers two distinct areas of the course: potential flow theory and heat exchangers. Make sure you attempt the questions from both halves of the sheet. 1  2A4A 2024 Part A: Applied Fluid Mechanics: Potential flow Lectures: Michaelmas Term Reading: B.S. Massey, Mechanics of Fluids, 7th edition, 1998 A. Potential flow Question 1 The change in stream function, Ψ, between two points in a steady, two- dimensional, incompressible, irrotational flow, is defined by the volume flow rate per unit depth passing between the points. Using the diagrams below find expressions for the components of velocity in cartesian and polar coordinates. 𝛹 + 𝑑𝛹 𝑦 𝑢𝑥 𝑑𝑦 𝑑𝑥 𝛹 −𝑢𝑦 𝑥 Cartesian Coordinate system Polar coordinate system 2  2A4A 2024 Question 2 a) Construct an accurate diagram of 8 consecutive integer-valued 𝑚𝜃 streamlines for a source (𝛹1 = ), of strength m = 8 centred at 𝑥 = 2𝜋 1, 𝑦 = 0. (Remember that the absolute value of stream function is arbitrary, as is the position chosen for the lowest assigned value. Thus, you may find it convenient to start by setting 𝜃 = 0 on the +ve 𝑥-axis, and later consider whether this should be adjusted). b) Construct on the same diagram (using a different line-type or colour) 9 streamlines with integer values from 0 to 8 for the uniform flow 𝜓2 = 4𝑥. c) By summing the stream functions at the intersections of the streamlines in the diagram, construct, again using a different line-type or colour) some streamlines for the flow 𝜓 = 4𝑥 + 8𝜃/2𝜋. d) Consider from the constructed streamlines whether there might be a stagnation (zero velocity) point in the combined flow field. Add velocity vectors due to the source (𝜓1 = 8𝜃/2𝜋) and the uniform flow (𝜓2 = 4𝑥) to find the precise location of this point. e) The streamline through the stagnation point is a dividing streamline which separates the stream fluid from the source fluid. Identify this streamline on your diagram and comment on what practical solid body this shape might represent. 3  2A4A 2024 Question 3 The stream function at point 𝑃(𝑥, 𝑦) resulting from a two-dimensional source of strength 𝑚 at (−𝑠, 0) and a matching sink (source of 𝑚𝑠𝑦 strength – 𝑚) at (𝑠, 0) may be written as 𝜓 = − (see notes). As 𝜋(𝑥 2 +𝑦 2 ) the separation 2s tends to zero the product m × s can be set to take a finite value. The stream function under these conditions is known as a doublet. In this question the doublet is superimposed upon a uniform stream of speed 𝑈 in the positive 𝑥 direction. The value for 𝑠 is very small and 𝑚 is very large, such that 𝑚𝑠 = 𝜋𝑈𝑎2 a) Show that the stream function of the combined flow in Cartesian coordinates is 𝑈 𝑎2 𝑦 𝜓 = 𝑈𝑦 − 2 𝑥 + 𝑦2 b) Transform this stream function into polar coordinates using 𝑥 = 𝑟 cos 𝜃, 𝑦 = 𝑟 sin 𝜃 and confirm that it represents the flow around a circular cylinder of radius 𝑎. c) Find general expressions for the radial and circumferential velocity in the flow field, and evaluate these at r = a. d) Use Bernoulli’s equation, and the velocities at the surface of the cylinder to express the surface pressure coefficient on the cylinder 4  2A4A 2024 𝑝 − 𝑝∞ 𝐶𝑝 = 1 2 𝜌𝑈 2 as a function of angular position 𝜃 from the upstream stagnation point and sketch 𝐶𝑝 between 𝜃 = 0 → 𝜋. e) How and why does the pressure distribution in a real flow differ from that predicted by potential flow theory? Question 4 The stream function of a circulation of strength Γ is defined as 𝜓 = Γ ln 𝑟 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. To make the stream function 𝜓 = 0 at 𝑟 = 𝑎, this is 2π Γ 𝑟 written as 𝜓 = ln. At large radius the stream function of a doublet 2π 𝑎 tends to zero. Thus, if we superimpose the stream function for the circulation on that of question 3 and consider the case of 𝑟 ≫ 𝑎, the combined stream function may be approximated as Γ 𝑟 𝜓 ≈ 𝑈𝑦 + ln 2π 𝑎 a) Write down the radial and circumferential components of velocity for this stream function. b) Apply the momentum equation using a cylindrical control volume of arbitrary radius, 𝑟 , where 𝑟 ≫ 𝑎, to find the force on the control volume perpendicular to freestream and confirm the Kutta-Joukowsky theorem i.e. 𝑙 = 𝜌𝑈Γ. [Hint: In addition to pressure forces on the control volume you will need to consider the momentum flux resulting from non-zero radial velocities]. 5  2A4A 2024 PART B Heat Exchangers Question 5 A heat exchanger is required to cool a flow of 1 kg s-1 of oil, specific heat 2 kJ kg-1 K-1, from 100°C to 60°C. The coolant is water, which enters the heat exchanger at 20°C at a flow rate of 0.5 kg s-1. The overall heat transfer coefficient is 1100 Wm-2 K-1. (a) For both counter-flow and co-current flow arrangements, (i) Calculate the water outlet temperature. (ii) Sketch the variation of the temperatures of the oil and water along the heat exchanger. (iii) Calculate the heat transfer area. (iv) Calculate the effectiveness. (b) For both arrangements (if appropriate) specify the heat exchanger required if the flow rate of water available is only 0.4 kg s-1. 6  2A4A 2024 Question 6 An air supply at ṁ = 5 kg s-1 is heated by passing it through a shell-and- tube heat exchanger. The tubes have an internal diameter d = 25 mm and condensing steam maintains the tube walls at essentially 100 °C. It is required to heat the air from Tin = 10 °C to Tout = 80 °C whilst dissipating 750 W in pumping. Take friction factor, f = 0.079 Re-0.25 and Stanton Number, St = 0.046 Re-0.25. a) Sketch a temperature position diagram and calculate the log mean temperature difference. b) If there are n tubes, write (i) the energy balance equation for a single tube and (ii) obtain an expression for the length of the pipe by considering the pressure drop Δp and the pumping power, P. (NB: P = Δp Q where Q is volumetric flow rate). c) Combine these results and use continuity to show that 2 8𝑓𝑚̇3 (𝑇𝑜𝑢𝑡 − 𝑇𝑖𝑛 ) 𝑛 = 2 4 2 𝜋 𝑑 𝜌 St Δ𝑇LM 𝑃 d) Evaluate the density of the air at the mean bulk flow temperature and a pressure of 101.3 kPa. Estimate the number of tubes required, and their length. 7  2A4A 2024 Question 7 Heat Exchanger Effectiveness – NTU analysis. (a) Plot effectiveness () vs number of transfer units (NTU) for co-current and counter current heat exchangers for the case of capacity rate ratios C = 0 and 1, all on the same axes in the range NTU = 0 - 6. Explain any difference in the asymptotic values of the effectiveness seen. (b) 1.5 kg s-1 of steam is to be condensed to saturated liquid at constant pressure by passing it over the tubes of a shell-and-tube heat exchanger. The steam enters the shell side of the counter-current heat exchanger with dryness fraction = 0.9 at 80 oC. (i) Find the total rate of heat transfer from the steam. Liquid water with an inlet temperature of Tc,in = 20 oC is passed through the heat exchanger tubes. The condenser contains 124, 10 mm diameter, thin-walled tubes (of negligible thermal resistance). The total water flow rate through the tubes is 15 kg s-1. The average heat transfer coefficient associated with condensation on the outer surface of the tubes may be taken as 5000 W m-2 K-1. For the water in the tubes take Cp = 4180 J kg-1 K-1,  = 0.001 kg m-1s-1, k = 0.61 Wm-1 K-1 and Pr = 6.85. (Check you know where to find these values in HLT). (ii) Find the effectiveness of the heat exchanger and use an appropriate graph from part (a) to find the number of transfer units, NTU. [Hint: think carefully about the appropriate value of capacity rate ratio for this question]. (iii) Find the heat transfer coefficient inside the tubes, and hence both the overall heat transfer coefficient, U, and the length of each heat exchanger tube. 8  2A4A 2024 Question 8 The CO2 that is circulated in a gas cooled nuclear reactor enters a steam raising unit at 550 °C, where steam is generated in a once-through boiler at a pressure of 150 bar superheated to 500°C. After being expanded to a pressure of 0.05 bar, the water is condensed to saturated liquid state and returned, via a pump, to the boiler. The temperature difference at the pinch-point is 30 K. Take cp for CO2 to be 1.01 kJ kg-1 K-1 at all temperatures. (a) Sketch the process block diagram, and temperature-enthalpy diagram across the boiler. (b) Hence, calculate: (i) the ratio of mass flow rates of water substance to that of CO2, (ii) the temperature of CO2 as it leaves the steam raising unit. (c) The boiler generates 330 kg s-1 of steam. Estimate the total heat transfer area in the evaporator section of the boiler. The overall heat transfer coefficient in the evaporator has a constant value of 160 W m-2 K-1. (d) What is the effect of increasing the temperature difference at the pinch point on the answers to (b) and (c)? 9  2A4A 2024 Some Answers Q1 Bookwork Q2. d)Stagnation Pt lies at (0, 1/𝜋) Q3. 𝑎2 a) 𝜓 = 𝜓1 + 𝜓2 = 𝑈𝑦 (1 − ) 𝑥 2 +𝑦 2 c) 𝑢𝑟 |𝑟=𝑎 = 𝑈 cos 𝜃 (1 − 𝑎2 /𝑟 2 )|𝑟=𝑎 = 0 d) 𝐶𝑝 = 1 − 4 sin2 𝜃 Q4 Γ a) 𝑢𝜃 − 𝑈 sin 𝜃 − , 𝑢𝑟 = 𝑈 cos 𝜃 2𝜋𝑟 b) Note the lift on the object which generates the circulation is equal and opposite to the force on the control volume Q5. (a) 58.3°C (both), 1.8 m2, 3.6 m2, ε = 0.5 (both) (b) 2.0 m2 , counter-flow, ε = 0.6 Q6. (a) 46.5 K 𝑃𝑑 (b) 𝐿 = (where u is the flow velocity) 𝑚̇ 2𝑓𝑢2 (d) 1.11 kg/m3, 852, 2.3 m Q7. (b) (i) 3110.13kW (ii) 0.80, 1.61 (iii) 6784 W / m2 K, 2878 W / m2 K, 9.00 m. Q8. 0.106, 219 °C, 29670 m2. 10