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Q1: Problem 2 [10 points] Consider a specially designed capillary tube, sealed at both ends, with variable diameter, as shown in the illustration below. This capillary is used for various experiments on the ISS (International Space Station). As this capillary floats motionless in the ISS cabin, will the air bubble inside the capillary move? See Answer
Q2: 5. Transient behavior of a stirred tank-E. The well-stirred tank of volume V = 2 m3 shown in Fig. P2.5 is initially filled with brine, in which the initial concentration of sodium chloride at t = 0 is co = 1 kg/m³. Subsequently, a flow rate of Q =0.01 m /s of pure water is fed steadily to the tank, and the same flow rate of brine leaves the tank through a drain. Derive an expression for the subsequent concentration of sodium chloride c in terms of co, t, Q, and V. Make a sketch of c versus t and label the main features. How long (minutes and seconds)will it take for the concentration of sodium chloride to fall to a final value of Cf = 0.0001 kg/m³? See Answer
Q3:1. Rouse' Law and the SPM profile equation (40pts): The Rouse' Law approach to suspended particulate matter (SPM) conservation assumes that SPM concentration is conserved locally, in each vertical column of water; i.e., SPM concentration may move up and down in the water col- umn (depending on flow conditions), but horizontal transport of SPM has no effect on the con- centration profile, even though individual particles will travel more or less with the flow and the concentration profile (once known) is used to calculate horizontal transport. This Rouse' Law ap- proach is the basis of most practical calculations of suspended load, except in some 3-D numeri- cal models, where the total derivative (local time variations plus horizontal advection) is consid- ered, also. The specific form of SPM conservation used in Rouse' law is (after Reynold's averag- ing and simplification using physical reasoning): a ac =W, DC + 2 (KBC) Ĉz cz 0=W₁ Here, C is SPM concentration, and Kc is a vertical diffusivity for SPM. (a) Derive the law for SPM conservation (15 pts): ac ac ac ôt ax (2) where no Reynolds' averaging has been done so far. There is a summary of the derivation in the first file on the equations of motion, after the discus- sion of mass conservation. You can follow the approach mentioned there, by considering the flux of suspended sediment. That is, define SPM as a volume concentration C (e.g., μ-liter SPM/liter water), and then define fluxes in and out of a control volume dVol = dx dy dz. Note that the flux you are defining is Cxpxvelocity and you are looking at the time-change of Cxp in volume dVol, so you are going to use a control volume approach to define: dxdydz = 2(Cp) dvol +U. +V ++ (w-ws) a(pCV) a(Cp) ôt ôt dz (3) To do this, you will have to assume that the sediment velocity vector Us = {u, v, w - Ws), where (u, v, w) is the water velocity. The SPM settling velocity Ws is NOT a function of {x,y,z,t) and does not have turbulent fluctuations - it is a constant. (b) Reynold's average (2) (which is your result from (a)) to get (10 pts): ac ac ac +u. +V +(w-ws) ac dz (1) = ac côz (4) (c) Now reduce the global expression of SPM conservation in (4) to the form needed for Rouse' Law (1), specifying each necessary assumption (I've already told you most of what you need to know.); 10 pts d) Explain why horizontal mixing is neglected in (4); 5 pts. 1See Answer
Q4:3) A problem on SPM profiles (40 pts): Consider a flow 12m deep with U- of 0.05 m/s and zo = 10 mm and Zref = 0.1 m (Zref is the location where the reference value of C is measured). A deep, rapidly flowing river like this would like have a continuum of sediment sizes that would have to be very simply "binned" for any sediment transport calculation. We will consider only three "size classes" (as they are called): silt d₁ = 25 µm, fine sand d₂ = 250 µm, and coarse sand d3 = 1000 μm (assume Corey shape factor =0.7), and information of the sort that one might be able obtain in a real system. (a) Find Ws, Ws/U., and the Rouse number P for each size class; 10 pts. (b) Assume that a log-layer velocity profile is adequate to describe the flow. What is that veloc- ity profile? 5 pts (c) What is the corresponding parabolic eddy diffusivity profile? 5 pts (d) Using the Rouse solution for the SPM profile in class notes (the solution applicable in a log- boundary layer), calculate for each size class the ratios: (Clz=10m/C|z=zref), (C|z= 1m/C|z=zref), and (C|z=0.5m/C|z=zref). If Clz- zref is the same for each of the three size classes, where do you expect the bulk of the sediment transport to occur for each size class? You don't need to know an actual value of Cz=zref to answer the question. But when working with observations, you want to have data (z= zref) as close to the bed as possible, to get good results. Otherwise, you'll miss the larger sizes; 20 pts 2See Answer
Q5:CHE 3333 Introduction to Transport Phenomena Fall 2023 Problem 1. (3 pts) A viscous film drains uniformly down the side of a vertical rod of radius R. You may assume that the flow is fully developed and that the liquid forms a film with a constant thickness of h. You may also neglect the shear resistance due to the atmosphere. Use the shell balance approach to determine the velocity distribution in the film.See Answer
Q6:Problem 2. (3 pts) Benzene, which is an incompressible Newtonian fluid, flows steadily and continuously at 100 °F through a 3,000 ft horizontal, 4" schedule 40 pipe. The pressure drop across the pipe under these conditions is 2 psi. You may assume fully developed, laminar flow. Use the shell balance approach to find the volumetric flow rate through the pipe in gallons per minute (gpm).See Answer
Q7:1. An infinitely long cylinder with a diameter of 10 cm is filled with a fluid. At x=0 and at t=0, a delta function mass input of 2.0 gm CO₂. Assume 25°C. Write governing equation, initial and boundary conditions and solution. Find the time to reach a concentration of 2 ppm at x=100 cm for (i) Molecular diffusion in air [2 pts]. Use D=0.164 cm²/s. [Be careful about converting mass/volume to mass/mass for air since 1 mg/l is not equal to 1 ppm in air! You will need the density of air: 1180 g/m³] (ii) Molecular diffusion in water [2 pts]. Use D=2E-5 cm²/s. (iii) Turbulent diffusion in water [2 pts]. As we will derive later, substitute a value of 1 cm²/s for D as a reflection of uniformly generated (isotropic and homogeneous) turbulence in water.See Answer
Q8:2. An infinitely long cylinder is filled with pure water on the right side of the partition, and water with an initial concentration of 100 ppm of dissolved CO₂ on the left. At time t=0 s, the partition is removed. Write governing equation, initial and boundary conditions and solution. (i) How long will it take the concentration of CO₂ at a point 50 cm to the right of the partition to reach 10 ppm? [2 pts] (ii) If the fluid was turbulent, how would that affect your answer above? As we will derive later, substitute a value of 1 cm²/s for D as a reflection of uniformly generated (isotropic and homogeneous) turbulence in water. How long will it now take to reach a CO₂ concentration of 10 ppm 50 cm from the partition? [2 pts]See Answer
Q9:3. Compare the result of problem 1 with the following: The cylinder has the mass input of 2.0 gm CO₂ added over a width of 5 cm (not at a point). Compare the results of this solution to Problem 1 for part (iii) for a D=1 cm²/s. First you must derive the solution using superposition (see similar derivation in the lecture notes). [5 pts]See Answer
Q10:4. Compare the result of problem 1(iii) with a numerical approach. Compute 2 cases for the numerical approach: Divide the grid into (a) 5 cm grid spaces and (b) 10 cm grid spaces and solve the governing equation using the numerical approach. Note that you will have to choose a numerical time step to ensure numerical "stability". Make sure you choose a value that Ax² satisfies this relationship: At <. Compare the numerical schemes with two different grids with the analytical solution in 1(iii) by plotting C vs x for each solution at t=100 s and comment on how well the numerical solution matches the analytical solution. [5 pts]See Answer
Q11:1. Molecular diffusion in a bounded medium [12 pts] An instantaneous point of strength M is located at the position (0, 0, h) above a horizontal plane (z = 0) that forms a reflective boundary. With a uniform advective velocity field (U, 0, 0) and molecular diffusivity D, (1) determine the solution for the resulting concentration distribution, and (2) by differentiation normal to the boundary show that the normal diffusive flux is indeed zero. [2 pts each] Assuming an additional horizontal boundary, that behaves as a perfect absorber for the substance under consideration, is located at z = H (where H>h), determine the solution. [3 pts] Assume a 1-D governing equation for vertical diffusion in a vertical tube: a²c Әс at = D azz assuming homogeneity in the x-y plane. Assume the tube has a length of 100 cm and there is an injection of mass in the middle of the tube (z=50 cm) of 10 g. Assume D is 0.001 m²/s and diameter of the tube is 10 cm. Perform 2 simulations with 2 different boundary conditions at z=0 cm and z=100 cm: (1) a concentration of zero, and (2) a no flux boundary condition. Use the numerical approach with a grid spacing of 5 cm to solve for the vertical distribution of mass after 60 s (i.e., plot C vs z at t-60 s) for (1) and (2). Choose your time step based on this criterion: At < 0.44² [5 pts] a. b. C.See Answer
Q12:1. Molecular diffusion in a bounded medium [12 pts] An instantaneous point of strength M is located at the position (0, 0, h) above a horizontal plane (z = 0) that forms a reflective boundary. With a uniform advective velocity field (U, 0, 0) and molecular diffusivity D, (1) determine the solution for the resulting concentration distribution, and (2) by differentiation normal to the boundary show that the normal diffusive flux is indeed zero. [2 pts each] Assuming an additional horizontal boundary, that behaves as a perfect absorber for the substance under consideration, is located at z = H (where H>h), determine the solution. [3 pts] Assume a 1-D governing equation for vertical diffusion in a vertical tube: дс a²c at = D assuming homogeneity in the x-y plane. Assume the tube has a əz² length of 100 cm and there is an injection of mass in the middle of the tube (z=50 cm) of 10 g. Assume D is 0.001 m²/s and diameter of the tube is 10 cm. Perform 2 simulations with 2 different boundary conditions at z=0 cm and z=100 cm: (1) a concentration of zero, and (2) a no flux boundary condition. Use the numerical approach with a grid spacing of 5 cm to solve for the vertical distribution of mass after 60 s (i.e., plot C vs z at t=60 s) for (1) and (2). Choose your time step based on this criterion: At < 0.44² [5 pts] a. b.See Answer
Q13:ase make space Problem 3: Numerical Solutions to Initial Value Problems (25 Points) Cylindrical cookies (radius: 1 cm and height 0.5 cm) are cooled on a conveyer belt after exiting an oven. The cookies enter the conveyer belt at 200 C. The goal is for the cookies to exit the conveyer belt at 40 C. The heat transfer coefficient at the surface of the coolie is 25 W/m²-K. The properties of the cookies are: C, (specific heat): 1000 J/kg-K K (thermal conductivity) = 1 W/m-K p (density) = 2000 kg/m³. Your goal is to determine the time taken for the cookies to cool to 40 C using numerical techniques./na. Write down (you don't need to solve this) the governing differential equation and initial condition that determines the temperature variation of the cookie with time (déjà vu!). (5 pts) (Please make space here) b. Using the Euler OR 2nd order Runge Kutta methods (you may choose any method of your choice Heun, Mid-Point, Ralston) and time step-sizes of your choosing, estimate the time-taken for the cookie to cool down to 40 C. Fill in your values in the table below i.e., you will be reducing your time-step size until you get a numerically converged solution (5 points) You don't need to use all the rows in the table if you have arrived at the numerically "correct" answer. On the other hand, please feel free to add more rows if it has taken you additional time-steps trials to arrive at the "correct" answer. Time-step size (sec/min/hrs) Time for the cookie to reach 40 C c. Choosing any time-step size and initial temperature of your choosing, show a sample step-by-step calculation (hand calculation) that demonstrates how you used the Euler/Runge Kutta methods to predict temperature at a future time. You only need to show this for only one time-step. Please show the calculations clearly to enable me to replicate your results (10 points)/nSample calculation Initial temperature (at the beginning of the time-step): Time-step size: Calculation of slope or slopes (k₁ and k₂ if using Runge-Kutta Method) (Please make space here) Temperature at the end of the time-step (Please make space here) d. If we do not have an exact analytical solution for this scenario, how did you ensure that you arrived at the numerically correct? (5 points) (Please make space here)See Answer
Q14:Problem 4: Numerical Methods (Shooting Method) (25 Points) Working for a company called Fin Design Inc., you are asked to design a cylindrical fin to increase the heat transfer rate from a pipe as shown in the figure below. Temperature of ambient air = Ta The relevant values in the figure are: R₁ = 0.8 in R₁= 1.0 in T,= 100 F Tv = 300 F R₁ Temperature T = T at r = Ro/nThe equation governing the temperature variation within the fin is given by: d²T 1 dr -4T=0 dr² r dr You may assume the tip of the fin is at the temperature of the surrounding air. Your goal is to determine the rate of heat transfer through this fin but for this you need to obtain the temperature distribution within the fin first. a. Dividing your fin into 4 equally spaced segments, use the shooting method to get the temperature profile within the fin. Fill in your values in the table below (if you used Goal seek/solver and obtained the correct value of w=dT/dr in your first try...that is OK!) (10 points) Estimate of slope (w = dT/dx) r (inches) 0.8 0.85 0.9 0.95 1.0 Fill in 1 estimate of "(w=dT/dr)" here T(r) - 1" estimate Fill in 2nd estimate of "(w=dT/dr)" here T(r) -2nd estimate Fill in 3rd estimate of "(w=dT/dr)" here T(r)-3rd estimate Fill in 4th estimate of "(w=dT/dr)" here T(r)-4th estimate b. Show a sample step-by-step calculation for any one of the columns that you report in the tables above (i.e., choosing any "r" location and any T (r) of your choosing, show how you used the shooting method to predict the T at the next spatial location - i.e., the number in the row below it). Please show the calculations clearly to enable me to replicate your results (10 points) Temperature at the first spatial location: Spatial-step size: Calculation of the slopes (dT/dr and dw/dr) at that spatial location:/n(Please make space here) Temperature at the next spatial location: (Please make space here) c. How did you ensure (or why do you feel) that you have arrived at the numerically correct answer? (5 points)See Answer
Q15:Problem 1 Water at 20°C is flowing down a vertical wall with Re = 10. Calculate (a) the flow rate, in gallons per hour per foot of wall length, and (b) the film thickness in inches.See Answer
Q16:Problem 2 The "Flow of Two Adjacent Immiscible Fluids" was solved in Class. However, we stopped short of actually evaluating the four integration constants. Using the four boundary conditions shown in the lecture slides, evaluate the four constants and show that the momentum flux and velocity profiles reduce to the answer shown on the slides.See Answer
Q17:Problem 3 Fluid is flowing through a narrow slit as shown below. Assuming no pressure differential between the inlet and outlet, derive: 1. An expression for velocity variation (along the x-direction only) within the slit 2. Calculate the ratios of average velocity to maximum velocity 3. Determine the mass flow rate within the slit. Fluid in Fluid out Flow through a slit, with B<< W<< LSee Answer
Q18:Problem 4 Repeat problem 3, assuming that the wall at x =B is moving in the positive z direction with a steady velocity of vo.See Answer
Q19:Q1 (2 points) Given extensional flow u = ÉxX, Uy = ÉyY₁ U₂ = ₂z, where x, y, and & are constant extensional rate along x, y and z axis, (a) please determine the relationship between Ex, y, and ₂ if flow is incompressible; (b) with = 1.0 s1, please calculate the time for a fluid element to extend 10 times of its original dimension along z axis.See Answer
Q20:Q2 (1.5 points) Given the viscosity μ-100 kPa.s and the modulus G-10 MPa of corn starch solution, please determine the critical time scale for the solution to have solid-like behaviors.See Answer

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