5) Determine the equilibrium temperature distribution for a one-dimensional rod with constant thermal properties with the following sources and boundary conditions:
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- Solve the one-dimensional Laplace equation for the equilibrium temperature distribution in a finite rod of length L for the following boundary conditions: 1. u(0) = -2 and u(L) = 2 2. u(0) = A and u(L) + ux (L) = 0 Please also draw a sketch for each boundary condition.Solve the one-dimensional Laplace equation for the equilibrium temperature distribution in a finite rod for the following boundary conditions: (a) u(0) = A and ux(L) = 0, Please, also draw a quick sketch for each boundary condition. If you need an initial condition for any reason you may assume that u(x, 0) = f(x).A mass balance equation describes the transient distribution of mass at every point in space. It is used extensively in many areas of chemical engineering just like environmental engineering. To monitor the pollutants in a well-mixed lake, a mass balance of the following equation is used W-kvvc c = Q = 1 x 105 m³/yr, V = 1 x 106 m³, and Parameter values are: W = 1 x 106 g/yr, k = 0.25 m0.5 / g0.5 / yr. Employ an initial guess of cof 4 and 5 g/m³. Perform three iterations and determine the percent true error after the third iteration. The root can be located with false position method.
- Solve for the equilibrium temperature distribution using the 2D Laplace equation on an L x H sized rectangular domain with the following boundary conditions: 1. Left: u(0,y) = f(y) (fixed temperature) 2. Bottom: u₂(x,0) = 0 (insulating) 3. Top: u₂(x, H) = 0 (insulating) 4. Right: u(L, y) = 0 (zero temperature) Solve for a general boundary temperature f(y). Also solve for the particular temperature distribution f(y) = sin(4Ty/H). u(0,y)-f(y) U₂, (x,H) = 0 7² - 0 u(L, y) - 0 u₂(x,0) = 0 Without too much extra work, tell me how this solution would change if we also made the right boundary condition insulating?Solve the one-dimensional Laplace equation for the equilibrium temperature distribution in a finite rod for the following boundary conditions: (a)u₂ (0) = a and u(L) = B, (b) u(0) + ux(0) = 0 and u(L) = B. Please, also draw a quick sketch for each boundary condition. If you need an initial condition for any reason you may assume that u(x,0) = f(x).A two-dimensional rectangular plate is subjected to prescribed boundary conditions, T1 = 50°C, T2 = 140°C. The temperature distribution equation, derived by applying separation of variable methods to a two-dimensional conduction problem for a thin rectangular plate or long rectangular rod, is as follows. (-1)*+1) + 1 -sin L sinh (nty/L) sinh (naW/L) nAX 0(x, y) = = Σ n n=1 Using this expression, calculate the temperature at the point (x,y) = (0.75, 0.5) by considering the first five nonzero terms of the infinite series that must be evaluated. Assume that L = 1.5 m. у (m) T2 1 T = 50°C- T = 50°C →x (m) L L -T = 50°C
- 4. Let y' = 4y – y². Find all equilibria and use the local stability criterion to determine if each is locally stable or unstable.Consider the cross section of a long rectangular metallic plate where the boundaries are subject to three different temperatures in degree Celsius, as shown in figure below. Engineers are interested in knowing the temperature distribution inside the plate in a specific period of time so they can determine the internal thermal stress. Assuming the boundary temperatures are held constant during that specific period of time, the temperature inside the plate will reach certain equilibrium after some time has passed. Finding this equilibrium temperature distribution at different points on the plate is desirable, but extremely difficult. However, one can consider a few points on the plate and approximate the temperature of these points. This approximation can be done using the mean value approach (the temperature will be approximated by averaging the 4 adjacent temperature, as we have done in class). a) b) 20 20 32 Metal Plate 24 32 24 24 24 What are the temperatures at x1 = °C, x2 = °C, X3…2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = = x²ex. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of vand x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mv = F) (iii) Assume the particle starts in x0 = 0 with positive initial velocity vo > 0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ô, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 ༧(2) vo m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > v. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the…
- 2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = x²e-x (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of v and x. Show that dE/dt =0, i.e., the total energy is constant during motion. (hint: use the equation of motion mi = F) (iii) Assume the particle starts in x = 0 with positive initial velocity vo >0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo> û, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 v(2) = ਪੰਨੇ m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > û. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the…2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = x²e¯x. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of v and x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mi = (iii) Assume the particle starts in xo energy Eo of the particle. Using (ii), with = F) O with positive initial velocity vo > 0. Find the initial show that the particle reaches x = 2 only if vo > û, 8e-2 = " m and in this case the particle's velocity in x = 2 is 8e-2 v(2): = v m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > û. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the…A mass m moves along the x-axis subject to an attractive force given by 19mx/2 and a retarding force given by , where x is its distance from the origin and is a constant. A driving force given by , where A is a constant, is applied to the particle along the x-axis. Write down the equation of motion. What value of results in steady-state oscillations about the origin with maximum amplitude? What is the maximum amplitude? what is the Q value?
