PROBLEM 2

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?

Molten chocolate is a Bingham fluid that flows through an 8 cm ID, 10 m long horizontal pipe with a pressure difference of 12 kPa. Find the volumetric flow rate in L/s, closest to: A) 0.323 B) 0.11323 C) 0.00113 D) 0.113 E) 0.03223

In polymer extrusion processes, a viscous polymer of viscosity u is forced to flowsteadily from left to right (distance L) in the annular area between two fixedconcentric cylinders by applying a pressure difference Pout - Pin. The inner cylinderis solid, whereas the outer one is hollow; their radii are R1 and R2, respectively.The problem, which could occur in the extrusion of plastic tubes, is to find thevelocity profile in the annular space and the total volumetric flow rate Q. Note thatcylindrical coordinates are now involved. (a) Giving reasons, simplify the continuity equation at steady state using cylindricalcoordinates \frac{\partial \rho}{\partial t}+\frac{1}{r} \frac{\partial}{\partial r}\left(\rho r v_{r}\right)+\frac{1}{r} \frac{\partial}{\partial \theta}\left(\rho v_{\theta}\right)+\frac{\partial}{\partial z}\left(\rho v_{z}\right)=0 (b) Giving reasons, simplify the Navier-Stokes equations for the velocity component which is not zero. \rho\left(\frac{\partial v_{r}}{\partial t}+v_{r} \frac{\partial v_{r}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{r}}{\partial \theta}+v_{z} \frac{\partial v_{r}}{\partial z}-\frac{v_{\theta}^{2}}{r}\right)=-\frac{\partial p}{\partial r}+\mu\left[\frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r}\left(r v_{r}\right)\right)+\frac{1}{r^{2}} \frac{\partial^{2} v_{r}}{\partial \theta^{2}}+\frac{\partial^{2} v_{r}}{\partial z^{2}}-\frac{2}{r^{2}} \frac{\partial v_{\theta}}{\partial \theta}\right]+\rho g_{r} \rho\left(\frac{\partial v_{\theta}}{\partial t}+v_{r} \frac{\partial v_{\theta}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{\theta}}{\partial \theta}+v_{z} \frac{\partial v_{\theta}}{\partial z}+\frac{v_{r} v_{\theta}}{r}\right)=-\frac{1}{r} \frac{\partial p}{\partial \theta}+\mu\left[\frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r}\left(r v_{\theta}\right)\right)+\frac{1}{r^{2}} \frac{\partial^{2} v_{\theta}}{\partial \theta^{2}}+\frac{\partial^{2} v_{\theta}}{\partial z^{2}}+\frac{2}{r^{2}} \frac{\partial v_{r}}{\partial \theta}\right]+\rho g_{\theta} \rho\left(\frac{\partial v_{z}}{\partial t}+v_{r} \frac{\partial v_{z}}{\partial r}+\frac{v_{\theta}}{r} \frac{\partial v_{z}}{\partial \theta}+v_{z} \frac{\partial v_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial v_{z}}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} v_{z}}{\partial \theta^{2}}+\frac{\partial^{2} v_{z}}{\partial z^{2}}\right]+\rho g_{z} (c) State the two boundary conditions needed to solve the simplified Navier-Stokes equations from (b). (d) Assuming negligible gravity effects, solve the equation derived in (b) subjected to the boundary conditions from (c) to show that the velocity profile of the viscous polymer flowing horizontally along the annulus is given by: u_{\mathrm{z}}=\frac{1}{4 \mu}\left(\frac{P_{O U T}-P_{I N}}{L}\right)\left[r^{2}-R_{1}^{2}+\frac{R_{1}^{2}-R_{2}^{2}}{\ln \left(\frac{R_{1}}{R_{2}}\right)} \ln \left(\frac{R_{1}}{r}\right)\right] (e) Derive an expression for the shear force (i.e., friction F,) arising from the shearing stress between the fluid flow (z-direction) and the external cylinder radial surface wall (r-direction).

4- A single-acting air compressor supplies 0.1 m³/s of air measured at, 273 K and 101.3 kN/m²which is compressed to 380 kN/m² from 101.3 kN/m². If the suction temperature is 289 K, the stroke is 0.25 m, and the speed is 4.0 Hz, what is the cylinder diameter? Assuming the cylinder clearance is 4 per cent and compression and re-expansion are isentropic (y = 1.4), what are the theoretical power requirements for the compression?

2. An intersection has a four-phase signal with the movements allowed in each phase, and corresponding analysis and saturation flow rates shown in Table below. Calculate the optimal cycle length, green, yellow, and all-red intervals for all four phases. Assume approach speed of 40 mi/h, intersection widths of 36 ft (NB and SB approaches) and 60 ft(EB and WB approaches). Assume left turns are of 60ft length and made at a speed of 20mi/h. Also check if the green intervals for through movements are sufficient for pedestrians to cross. Assume startup lost time = 2 sec, clearance lost time = 2 sec. Assume zero grade.No. of pedestrians crossing during an interval= 10, and crosswalk width (We) = 8ft.

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).

A continuous belt passes upward through a chemical bath at velocity Vo and picks up a film of liquid of thickness h, density p, and viscosity μ. Gravity tends to make the liquid drain down, butthe movement of the belt keeps the fluid from running off completely. Assume that the flow is a well-developed laminar flow with zero pressure gradient, and that the atmosphere produces no shear at the outer surface of the film. Use the shell-balance approach to (1) derive the governing differential equations. (2) State the boundary conditions for the systems. (3) Determine the velocity profile. Clearly list any assumptions needed. [DO NOT sketch the velocity profile.]

3. A large jug filled with water has three holes poked at the same time. Immediately, it begins leaking as shown in the figure below. Each fluid stream has a diameter of 0.15 inches., and the distance between each of the holes is 2 inches. Neglecting viscous effects, estimate the time it takes for water to stop draining from the first hole. Assume that the top hole is 2 inches beneath the water's surface when the holes are poked.

A continuous belt passes upward through a chemical bath at velocity Vo and picks up a film of liquid of thickness h, density p, and viscosity u. Gravity tends to make the liquid drain down, but the movement of the belt keeps the fluid from running off completely. Assume that the flow is a well-developed laminar flow with zero pressure gradient, and that the atmosphere produces no shear at the outer surface of the film. Use the shell-balance approach to (1) derive the governing differential equations. (2) State the-boundary conditions for the systems. (3)Determine the velocity profile. Clearly list any-assumptions needed. [DO NOT sketch the velocity profile.]

Species Accumulation and Transport within an Interface Accumulation and transport within an interface can be important for species that reside at phase boundaries, such as gases adsorbed on solids or surfactants at fluid-fluid interfaces. The objective is to derive more general interfacial conservation statements than those in Section 2.2. Consider a species that may be present at an interface or in either of the adjacent bulk phases. Let Cs and Ng be its surface concentration (moles m2) and surface flux (moles m¹s¹), respectively; note the differences in units from the corresponding three-dimensional quantities. The vector Ng is every- where tangent to the surface. (a) For part of an interface corresponding to surface S in Fig. A-2, state the macroscopic (integral) solute conservation equation. Assume that phase A is below S and phase B above it, such that n points toward the latter. Include the possibility of chemical reactions at the interface, trans- port to or from the bulk phases, and interfacial motion. The interface is not necessarily planar. (b) By reducing the integral equation of part (a) to a partial differential equation, show that at where N and C are the species flux and concentration, respectively, in a bulk phase. (Sub- scripts identifying the chemical species have been dropped for simplicity.) This result, which is valid instantaneously and locally, is equivalent to Eq. (5.2-2) of Edwards et al. (1991). How does it compare with what is obtained by applying Eq. (2.2-15) to a chemical species?