Questions and Answers of Fundamentals Of The Finite Element Method

In the above problem if the linear variation of temperature is replaced with a constant heat flux, determine the temperature and flow patterns.
Derive the energy balance equation for a rectangular fin of variable crosssection as shown in Figure 1.6. The fin is stationary and is attached to a hot heat source.
A closed plastic container used to serve coffee in a seminar room is made of two layers with an air gap placed between them. List all heat transfer processes associated with the cooling of the coffee
A square chip of size \(8 \mathrm{~mm}\) is mounted on a substrate with the top surface being exposed to a coolant flow at \(20^{\circ} \mathrm{C}\). All other surfaces of the chip are insulated. The
Consider a person standing in a room, which is at a temperature of \(21^{\circ} \mathrm{C}\). Determine the total heat rate from this person if the exposed surface area of the person is \(1.6
A thin metal plate has one large insulated bottom surface and the top large surface exposed to solar radiation at a rate of \(600 \mathrm{~W} / \mathrm{m}^{2}\). The surrounding air temperature is
A long, thin copper wire of radius \(r\) and length \(L\) has an electrical resistance of \(ho\) per unit length. The wire is initially kept at a room temperature of \(T_{a}\) and subjected to an
In a continuous casting machine, the billet moves at a rate of \(\mathrm{m} / \mathrm{s}\). The hot billet is exposed to an ambient temperature of \(T_{a}\). Set up an equation to find the
In a double pipe heat exchanger, hot fluid (mass flow \(M \mathrm{~kg} / \mathrm{s}\) and specific heat \(\mathrm{C} \mathrm{kJ} / \mathrm{kg}{ }^{\circ} \mathrm{C}\) ) flows inside a pipe and cold
Use the system analysis procedure described in this chapter and construct the discrete system for heat conduction through the composite wall shown in Figure 2.7. Also, from the following data,
The cross-section of an insulated pipe carrying a hot fluid is shown in Figure 2.8. The inner and outer radii of the pipe are \(r_{1}\) and \(r_{2}\) respectively. The thickness of the insulating
The pipe network used to circulate hot water in a domestic central heating arrangement is shown in Figure 2.9. The flow rate at the entrance is \(\mathrm{Qm}^{3} / \mathrm{s}\). Neglecting any loss
A schematic diagram of a counter flow heat exchanger is shown in Figure 2.10. The hot fluid enters the central, circular pipe from the left and exits at the right. The cooling fluid is circulated
A transient analysis is very important in the casting industry. In Figure 2.11, a simplified casting arrangement is shown (without a runner or raiser). The molten metal is poured into the mold and
Consider a \(0.6 \mathrm{~m}\) high and \(2 \mathrm{~m}\) wide double-pane window consisting of two \(4 \mathrm{~mm}\) thick layers of glass \(\left(k=0.80 \mathrm{~W} / \mathrm{m}^{\circ}
A simplified model can be applied to describe the steady-state temperature distribution through the core region, muscle region and skin region of the human body. The core region temperature
A composite wall consists of layers of aluminum, copper and steel. The steel external surface is \(350^{\circ} \mathrm{C}\) and the external surface of the aluminum is exposed to an ambient of
An incompressible fluid flows through a pipe network of circular pipes as shown in Figure 2.12. If \(0.1 \mathrm{~m}^{3} / \mathrm{s}\) of fluid enters and leaves the piping network, using a 4-node
Figure 2.13 shows a direct current circuit. The voltage at the output terminals are also shown in Figure 2.13. Calculate the voltage at each node and the current in each of the branches using the
A cross-section of a heat sink used in electronic cooling is shown in Figure 2.14. All the fins are of the same size. Calculate the heat dissipating capacity of the heat sink per unit length of heat
The details of a double pipe heat exchanger are given as: (a) cold fluid heat capacity rate \(C_{1}=1100 \mathrm{~W} /{ }^{\circ} \mathrm{C}\); (b) Hot fluid heat capacity rate \(C_{2}=734
Figure 2.15 shows an arrangement for cooling of an electronic equipment consisting of a number of printed circuit boards (PCB) enclosed in a box. Air is forced through the box by a fan. Select a
A one-dimensional linear element is used to approximate the temperature variation in a fin. The solution gives the temperature at two nodes of an element as \(100^{\circ} \mathrm{C}\) and
A one-dimensional quadratic element is used to approximate the temperature distribution in a long fin. The solution gives the temperature at three nodes as 100, 90, and \(80^{\circ} \mathrm{C}\) at
During the implementation of the finite element method, the evaluation of the integrals that contain shape functions and their derivatives is required. Evaluate the following integrals for a linear
Derive the shape functions for a one-dimensional linear element in which both the temperature and the heat fluxes should be continuously varying in the element. (Note that degrees of freedom for a
The solution for temperature distribution in a linear triangle gives the nodal temperature as \(T_{i}=200^{\circ} \mathrm{C}, T_{j}=180^{\circ} \mathrm{C}\) and \(T_{k}=160^{\circ} \mathrm{C}\). The
For a one-dimensional quadratic element evaluate the integrals. (Note: convert \(N_{i}, N_{j}\) and \(N_{k}\) to local coordinates and then integrate.)\[ \begin{equation*} \int_{l} N_{i} d l ;
The nodal values for a rectangular element is given as follows, \(x_{i}=0.25 \mathrm{~cm}\), \(y_{i}=0.20 \mathrm{~cm}, x_{j}=0.30 \mathrm{~cm}, y_{m}=0.25 \mathrm{~cm}, T_{i}=150^{\circ} \mathrm{C},
Calculate the shape functions for a six-noded rectangle shown in Figure 3.31. 4 5 3 cm 2 3 cm 3 3 cm
Evaluate the partial derivatives of shape functions at \(\zeta=1 / 4\) and \(\eta=1 / 2\) of a quadrilateral element shown in Figure 3.32, assuming the temperature is approximated by (a) bilinear,
Calculate the derivatives \(\partial N_{6} / \partial x\) and \(\partial N_{6} / \partial y\) at a point \((2,5)\) for a quadratic triangle element shown in Figure 3.33 using local coordinates. (1,1)
In a double pipe heat exchanger, hot fluid flows inside a pipe and cold fluid flows outside in the annular space. The heat exchange between the two fluids is given by the differential equations,
Calculate (using one, two and four elements) the temperature distribution and the heat dissipation capacity of a fin of length \(4 \mathrm{~cm}\) and cross-sectional dimensions of \(6 \mathrm{~mm}
A composite wall with three different layers, as shown in Figure 4.2 generates \(0.25 \mathrm{GW} / \mathrm{m}^{3}\) of heat. Using the relevant data given in Example 4.2.1, determine the temperature
An insulation system around a cylindrical pipe consists of two different layers. The first layer immediately on the outer surface of the pipe is made of glass wool and the second one is constructed
A solid cylinder of \(10 \mathrm{~cm}\) diameter generates \(0.3 \mathrm{GW} / \mathrm{m}^{3}\) of heat due to nuclear reaction. If the outside temperature is \(40^{\circ} \mathrm{C}\) and the heat
A circular fin of inner diameter \(20 \mathrm{~cm}\) and outer diameter of \(26 \mathrm{~cm}\) transfers heat from a small motor cycle engine. If the average engine surface temperature is
Consider a composite wall consisting of four different materials as shown in Figure 4.15. Assuming one-dimensional heat flow, determine the heat flow through the composite slab and the interfacial
Consider a composite wall, which has one linearly varying cross-sectional area as shown in Figure 4.16. Determine the heat flow and interfacial temperatures. Thickness \(=10 \mathrm{~cm}, k_{A}=200
A plane wall \(\left(k=20 \mathrm{~W} / \mathrm{m}{ }^{\circ} \mathrm{C}\right)\) of thickness \(40 \mathrm{~cm}\) has its outer surfaces maintained at \(30^{\circ} \mathrm{C}\). If there is uniform
A plane wall \(\left(k=10 \mathrm{~W} / \mathrm{m}^{\circ} \mathrm{C}\right)\) of thickness \(50 \mathrm{~cm}\) has its exterior surface subjected to convection environment of \(30^{\circ}
Calculate the outer wall surface temperature and the temperature distribution in a thick walled hollow cylinder when the inner wall temperature is \(120^{\circ} \mathrm{C}\) and the outer wall is
Calculate the surface temperature in a circular solid cylinder \((k=\) \(20 \mathrm{~W} / \mathrm{m}^{2} \mathrm{C}\) ) of radius \(30 \mathrm{~mm}\) with a volumetric heat generation of \(25
Consider a tapered fin of length \(5 \mathrm{~cm}\) dissipating heat to an ambient at \(30^{\circ} \mathrm{C}\). The heat transfer coefficient on the surface at the tip is \(100 \mathrm{~W} /
A square plate size \(100 \mathrm{~cm} \times 100 \mathrm{~cm}\) is subjected to an isothermal boundary condition of \(500^{\circ} \mathrm{C}\) on the top and to convection environment (on all the
If in Exercise 5.8.1, there is a uniform heat generation of \(2{\mathrm{~W} / \mathrm{cm}^{3}}^{3}\) exists, and a line source of \(5 \mathrm{~W} / \mathrm{cm}\) at a location of \((x=30
Repeat Exercise 5.8.1 using(a) one rectangle;(b) four rectangles.Data From Exercise 5.8.1 A square plate size \(100 \mathrm{~cm} \times 100 \mathrm{~cm}\) is subjected to an isothermal boundary
Repeat Exercise 5.8. using(a) one rectangle;(b) four rectangles.Data From Exercise 5.8.1 A square plate size \(100 \mathrm{~cm} \times 100 \mathrm{~cm}\) is subjected to an isothermal boundary
In Exercise 5.8.1, if the thickness increases uniformly from \(1 \mathrm{~cm}\) from the bottom edge to \(3 \mathrm{~cm}\) at the top edge, rework the problem with(a) two triangles;(b) eight
Calculate the stiffness matrix and loading vector for the axisymmetric element shown in Figure 5.19 with heat generation of \(G=1 \mathrm{~W} / \mathrm{cm}^{3}\), the heat transfer coefficient on the
An Internal Combustion (IC) engine cylinder is exposed to hot gases at \(1000^{\circ} \mathrm{C}\) on the inside wall with a heat transfer coefficient of \(25 \mathrm{~W} / \mathrm{m}^{2}
A large block of steel with a thermal conductivity of \(40 \mathrm{~W} / \mathrm{m}^{\circ} \mathrm{C}\) and a thermal diffusivity of \(1.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) is initially
A fin of length \(1 \mathrm{~cm}\) is initially at the ambient temperature of \(30^{\circ} \mathrm{C}\). If the base temperature is suddenly raised to a temperature of \(150{ }^{\circ} \mathrm{C}\)
A short aluminum cylinder \(2.5 \mathrm{~cm}\) in diameter and \(5 \mathrm{~cm}\) long is initially at a uniform temperature of \(100^{\circ} \mathrm{C}\). It is suddenly subjected to a convection
A plane wall of thickness \(4 \mathrm{~mm}\) has internal heat generation of \(25 \mathrm{MW} / \mathrm{m}^{3}\) with thermal properties of \(k=20 \mathrm{~W} / \mathrm{m}^{\circ} \mathrm{C}, p=8000
A stainless steel plate size \(2 \mathrm{~cm} \times 1 \mathrm{~cm}\) is surrounded by an insulating block as shown in Figure 6.14 and is initially at a uniform temperature of \(40^{\circ}
Using a differential control volume approach, derive a three-dimensional convection-diffusion equation for pollution transport in a river.
Following the derivation of the CG method for a convection-diffusion equation discussed in this chapter, derive the CG method for a convection-diffusion equation with a source term \(Q\).
Derive Navier-Stokes equations in cylindrical and spherical coordinates.
Reduce the incompressible Navier-Stokes equations to solve a onedimensional time-dependent convection heat transfer problem.
For natural convection problems, if \(\alpha\) is replaced by \(v\) in the nondimensional scaling, derive the new nondimensional form.
Calculate laminar flow and heat transfer from a hot cylinder at \(R e=40\) placed inside a rectangular channel (assume the size) using the CBSflow code. Assume buoyancy effect is negligible.
Write a program in any standard scientific language to calculate stream functions from a computed velocity field.
In this example, you are asked to make appropriate assumptions and model flow past the heat exchanger tubes as shown in Figure 7.54.A schematic diagram of a typical cross flow heat exchanger
In this exercise, you are asked to simulate the liquid flow through a liquid processing plant as shown in Figure 7.55.In the liquid processing industry, liquid is passed through several tanks as
A two-dimensional square enclosure (all solid walls) filled with air is subjected to a linearly varying temperature on one of its vertical walls (say \(T=\left(x_{2} / L\right) T_{\max }\), where