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Prove that the 1-D heat conduction equation \frac{\partial T}{\partial t}=\alpha \frac{\partial^{2} T}{\partial x} is a parabolic equation.

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Q3: Increase the inlet speed used in the previous question so that the Reynolds number becomes 5000, solve the problem again and provide new results. Before solving the problem initialize it,i.e. do not continue over the previous solution. Is there any unexpected behavior in this solution?If there is any, what is the cause of it? Provide the residual plot and discuss its behavior.

[10 pts.]Using Taylor's series, derive the first-order, forward-difference expression for dul dy.

Q2: Grid convergence is required for all CFD analysis. When grid size is reduced gradually, certain critical parameter gradually reaches convergence. In the present study, the drag coefficient can betaken as the critical parameter. Reduce the mesh size and solve the problem again, until grid convergence is achieved. Before solving the problem initialize it, i.e. do not continue over the previous solution.

For general design of open channels, briefly answer the following: a) Why do gradually varied flow calculations using the direct step method typically require that velocity between any 2 depths not vary by more than 10%? b) Why is determination of critical depth important? c) What is the most importantoverall design objective? For general design of open channels, briefly answer the following:

.Describe how non-linear terms in the Navier-Stokes equation are modeled in the finite difference method? Use the term u(dul əx) as your example. Comment on the accuracy of the approximation.

Q2 Consider the non-staggered tube bank arrangement as shown below. All the conditions are the same as that from periodic tube bank heat transfer tutorial except the geometry. The tube diameter's 1cm, You need: 1) identify a small region for the CFD simulation; 2) generate the appropriate geometry using ANSYS 3) generate the necessary mesh 4) follow the steps in the tutorial to perform CFD simulation of the heat transfer in a non-stagger tube bank 5) compare you results with that from the staggered tube bank and explain the difference in terms of heat transfer and pressure distribution around the cylinder. 6) show the pressure contours (show at least 3 rows and four columns of tube banks), 7) show the temperature contours 8) plot the streamlines 9) temperature and pressure variations along the vertical lines A-A and B-B

[10 pts.]Derive the second-order central difference approximation for the term du/ dy using Taylor's series.

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. For incompressible, Newtonian, laminar flow with heat transfer (but no reactions), list the name of the equations (just the name, no need to write down the full equations) to be solved,and state the number of equations and all the scalar unknowns to be solved for.

A 36-in diameter water supply line was installed 20 years ago on a slope of 0.1% and designed to flow nearly full (D/d = 0.95). If the Manning coefficient at time of installation was 0.015 and current pipe capacity is 15 cfs, answer the following: a) Calculate flow velocity immediately after installation b) Calculate current flow velocity and current value of Manning coefficient c) Briefly explain why a water supply line can be designed to flow safely whennearly full but a sewer or storm drain cannot

Q4: run the case of Re=5000 using transient solver. Monitor the lift force or lift coefficient of the cylinder. Estimate the oscillation frequency of the flow from the profile of life force/coefficient and compare it with the data in literature. Experiments show that the frequency, f, of this periodic flow (in Hz) can be expressed in terms of a non-dimensional parameter known as the Strouhal number, defined as: S t=\frac{f D}{U_{\infty}}

a)Describe why departure functions are used to calculate changes in thermodynamic properties(4)under non-ideal conditions. (b)Butane gas undergoes a change of state from an initial condition of 2 MPa and 160°C to 3.5MPa and 227°C. Using departure functions and the thermodynamic data below calculate thechange in enthalpy and entropy.(12) \text { The heat capacity at constant pressure is } c_{\mathrm{p}}=9.487+0.3313 T-1.108 \times 10^{-4} T^{2}-2.822 \times 10^{-9} T^{3} \text { in } units of J kg1 K, where T is the temperature in kelvin. The enthalpic and entropicdeparture functions for butane are given by: \text { At } 2 \mathrm{MPa} \text { and } 160^{\circ} \mathrm{C}, h-h^{\mathrm{ig}}=-2.4263 \mathrm{~kJ} / \mathrm{mol} \text { and } s-s^{\mathrm{i} 8}=-3.9507 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} \text { At } 3.5 \mathrm{MPa} \text { and } 227^{\circ} \mathrm{C}, h-h^{i 8}=-3.2693 \mathrm{~kJ} / \mathrm{mol} \text { and } s-s^{\mathrm{i} 8}=-4.7567 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} Using the Peng-Robinson equation of state, what is h- h for 56.11 g of 1-butene at 2 MPa and160°C?(c)(4) You might find the PREOS spreadsheet useful to answering part (c), but you are welcome to useother sources of information if you wish. You must describe how you went out about answeringthis question, which is simply assessing your ability to obtain information.

0.Outline the major differences between the explicit and implicit schemes used to solve difference equations. Which scheme is more stable? Are the implicit schemes always stable? Elaborate briefly.

28 ROR with Interpolation Ans. it = 8.75?

7. Using the von Neumann stability analysis, determine the stability requirement of the following FDE (finite-difference equation): where a > 0 is a constant.10

Consider a cylinder in the wind tunnel of the thermal fluid laboratory. The cylinder is about 10.5 inches long and has a diameter of 1.891 inch. The cross section of the duct is 10.5 inches wide and 15.75 inches high. The length of the test section is 42 inches and the cylinder is located approximately 16 inches downstream of the entrance to the test section. Now the three dimensional problem is simplified as two dimensional problem (Figure 1). The computational domain covers the whole duct. Q1: the Reynolds number is defined as ReD=PUD/u where D is the diameter of the cylinder, U is the inlet velocity. Set up the case that has Reynolds number of 25. Make sure that the convergence tolerances are IE-5. Mesh size 10 mm with level 3 refinement around the cylindrical obstacle and on the wall. Initialize the problem and solve it. Provide the following results: (a) Number of iterations needed for convergence. (b) Drag coefficient (c) Contour plot of velocity magnitude. (d) Pathline plot (e) x velocity profiles along the following five vertical lines: at the center of the cylinder (x=0),x= -50 mm, x= -26 mm, x = 50 mm, x= 400 mm, as approximately shown below.

Consider 1-D steady heat conduction in a composite plane wall (shown below) consisting of two layers A and B in perfect contact at the interface. The wall involves no heat generation.The nodal network of the wall consists of nodes 0 (at the left face), 1 (at the interface), and 2(at the right face) with a uniform nodal spacing of Ax. Using the energy balance approach,obtain the finite difference formulation of this problem for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 2) with an emissivity of & and surrounding temperature of Tsurr.

Specific energy is given below for a range of depths in a trapezoidal channel with a base width of 6 ft and a 2:1 side slope a) Calculate flow rate (cfs) in channel b) Calculate critical depth in channel c) Briefly explain how critical depth is used to measure flow rate in an open channel

A Cal Poly SLO student falls into a long rectangular concrete flood control channel that carries 35,000 cfs during a major storm. The channel is 50 ft wide with an average slope of 3.2%. Answer the following: a) Estimate how many seconds for student to travel 1 mile (5,280 ft) downstream b) Calculate Froude number and state whether a hydraulic jump is possible c) Estimate maximum water depth in channel at which a hydraulic jump could form d) Briefly explain why depth immediately after approximately equal downstream normal depth (i.e. tailwater depth) hydraulic jump should

[10 pts.]2. Consider steady one-dimensional heat conduction in a pin fin of constant diameter D, like the one shown below, with constant thermal conductivity k. The fin is losing heat by convection to the ambient air at T with a heat transfer coefficient of h. The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of Ax. Using the energy balance approach, obtain the finite difference formulation of this problem to determine T₁ and T₂ for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. Temperatures are in °C.

What is the significance of Burger's equation? Elaborate briefly.

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