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

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.

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.

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.

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.

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

[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.