## Yogesh Jaluria

## Chapter 4

## Numerical Modeling and Simulation - all with Video Answers

## Educators

Chapter Questions

The mass balance for three items $x, y$, and $z$ in a chemical reactor is governed by the following linear equations:

$$

\begin{aligned}

2.2 x+4.5 y+1.1 z & =11.14 \\

4.8 x+y+2.5 z & =-1.62 \\

-2.1 x-3.1 y+10.1 z & =15.57

\end{aligned}

$$

Solve this system of equations by the Gauss-Seidel method to obtain the values of the three items. You may arrange the equations in any appropriate order. Do you expect convergence? Justify your answer. The initial guess may be taken as $x=y=z=0.0$ or 1.0 .

Check back soon!

An industrial system has three products whose outputs are represented by $x, y$, and $z$. These are described by the following three equations:

$$

\begin{aligned}

1.8 x-3.1 y+7.6 z & =12.2 \\

4.8 x+6 y-1.1 z & =24.8 \\

3.3 x+1.7 y+0.9 z & =13.0

\end{aligned}

$$

(a) Give the block representation for each of these subsystems.

(b) Draw the information-flow diagram for the system.

(c) Set up this system of equations for an iterative solution by any appropriate method, starting with an initial guess of $x=y=z=0$.

(d) Show at least 5 iterative steps to obtain the solution to simulate the system.

Check back soon!

The mass balance for three items $a, b$, and $c$ in a reactor is given by the following linear equations:

$$

\begin{aligned}

4 a+2 b+2 c & =17 \\

a-5 b+c & =-5 \\

2 a+3 b-6 c & =-12

\end{aligned}

$$

Solve this system of equations by the Gauss-Seidel iteration method. The initial guess may be taken as $a=b=c=0.0$ or 1.0 .

Check back soon!

Solve the following set of linear equations by the Gauss-Seidel iteration method. The initial guess may be taken as 0.0 or 1.0 .

$$

\begin{aligned}

5 x+y+2 z & =17 \\

x+3 y+z & =8 \\

2 x+y+6 z & =23

\end{aligned}

$$

Vary the convergence parameter to ensure that results are independent of the value chosen.

Check back soon!

A firm produces four items, $x_1, x_2, x_3$, and $x_4$. A portion of the amount produced for each is used in the manufacture of the other items. The balance between the output and the production rate yields the equations

$$

\begin{aligned}

x_1+2 x_2+5 x_4 & =32 \\

3 x_1+2 x_2+6 x_3 & =36 \\

3 x_1+5 x_2+2 x_3+x_7 & =41 \\

2 x_2+10 x_3+8 x_4 & =58

\end{aligned}

$$

Solve these equations by the SOR method and determine the optimum value of the relaxation factor $\omega$. Obtain the production rates of the four items. Compare the number of iterations needed for convergence at the optimum $\omega$ with that for the Gauss-Seidel method $(\omega=1)$.

Check back soon!

Using the successive substitution and the Newton-Raphson methods, solve the following equation for the value of $x$, which is known to be real and positive:

$$

x^5=\left[10(10-x)^{0.5}-8\right]^3

$$

The equation may be recast in any appropriate form for the application of the methods. Compare the solution and the convergence of the numerical scheme in the two cases.

Sarah Parrigin

Numerade Educator

The solidification equation for casting in a mold at temperature $T_a$, considering energy storage in the solid, is obtained as

$$

C\left(T_m-T_a\right) / L \pi^{1 / 2}=\eta \exp \left(\eta^2\right) \operatorname{erf}(\eta)

$$

where $C$ is the material specific heat, $T_m$ is the melting point, $L$ is the latent heat, and $\eta=\delta /\left[2(\alpha \tau)^{1 / 2}\right], \delta$ being the interface location, as shown in Figure P4.7, $\alpha$ is the thermal diffusivity, and $\tau$ is the time. Take $\alpha=$ $10^{-5} \mathrm{~m}^2 / \mathrm{s}, L=110 \mathrm{~kJ} / \mathrm{kg}, T_m=925^{\circ} \mathrm{C}, T_a=25^{\circ} \mathrm{C}$, and $C=700 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ for the material being cast. Approximate the error function $\operatorname{erf}(\eta)=\eta$, for $0<\eta<1$, and $\operatorname{erf}(\eta)=1.0$ for $\eta>1$. Solve this equation for $\eta$ and calculate the interface location $\delta$ as a function of time $\tau$. What is the casting time for a $0.4-\mathrm{m}$-thick plate, with heat removal occurring on both sides of the plate?

(FIGURE CAN'T COPY)

Check back soon!

For the casting of a plate 10 cm thick, use the graphs presented in Example 4.8 to determine the total solidification times for the cases when the mold is 2 cm or 10 cm thick. Also, determine the time needed to solidify $75 \%$ of the plate. The heat transfer coefficient $h$ is given as $40 \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}$.

Check back soon!

A spherical casting of diameter 10 cm has a total solidification time (TST) of 5 min . Assuming Chvorinov's model, TST $=C(V / A)^2$, where $V$ is the volume, $A$ is the surface area, and $C$ is a constant, calculate the diameter of a long cylindrical runner with a TST of 12 min .

Check back soon!

The speed $V$ of a vehicle under the action of various forces is given by the equation

$$

5.0 \exp (V / 3)+2.5 V^2+2.0 V=20.5

$$

Compute the value of $V$, using any appropriate method. Justify your choice of method. Suggest one other method that could also have been used for this problem.

Manish Jain

Numerade Educator

The temperature $T$ of an electrically heated wire is obtained from its energy balance. If the energy input into the wire, per unit surface area, due to the electric current is $1000 \mathrm{~W} / \mathrm{m}^2$, the heat transfer coefficient $h$ is $10 \mathrm{~W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}\right)$, and the ambient temperature is 300 K , as shown in Figure P4.11, the resulting equation is obtained as

$$

1000=0.5 \times 5.67 \times 10^{-8} \times\left[T^4-(300)^4\right]+10 \times(T-300)

$$

Calculate the temperature of the wire by the secant method. Using this numerical simulation, determine the effect of the energy input on the temperature by varying the input by $\pm 200 \mathrm{~W} / \mathrm{m}^2$. Also, vary the ambient temperature by $\pm 50 \mathrm{~K}$ to determine its effect on the temperature. Do the results follow the expected physical trends?

(FIGURE CAN'T COPY)

A cylindrical container of diameter $D$ is placed in a stream of air and the energy transfer from its surface is measured as 100 W . The energy balance equation is obtained using correlations for the heat transfer coefficient as

$$

\left[\frac{60}{D} D^{0.466}+50\right] \pi D=100

$$

Find the diameter of the container using any root-solving method. Also, use this simulation to determine the diameter needed for losing a given amount of energy in the range $100 \pm 20 \mathrm{~W}$ by varying the heat lost.

Ajay Singhal

Numerade Educator

Use the bisection method to determine the root of the equation

$$

x\left[1-\exp \left(-\frac{10}{1+4 x}\right)\right]-1=0

$$

Mohamed Raafat Mohamed

Numerade Educator

Use the successive substitution method to determine the variable $v$ from the equation

$$

v=\left\{\left[\left(\frac{14-v}{72 * 10^{-6}}\right)^{0.5}-85\right] / 10.8\right\}^{0.65}

$$

Narayan Hari

Numerade Educator

Use Newton's method or the secant method to solve the equation

$$

\exp (x)-x^2=0

$$

Stanley Enemuo

Numerade Educator

Use Newton's method to find the real roots of the equation

$$

x^4-4 x^3+7 x^2-6 x+2=0

$$

Mohamed Raafat Mohamed

Numerade Educator

The root of the equation

$$

[\exp (-0.5 x)] x^{1.8}=1.2

$$

is to be obtained. It is given that a real root, which represents the location of the maximum heat flux, lies between 0 and 6.0 . Using any suitable method, find this root. Give reasons for your choice of method. What is the expected accuracy of the root you found?

James Kiss

Numerade Educator

The generation of two quantities, $F$ and $G$, in a chemical reactor is governed by the equations

$$

\begin{gathered}

2.0 F^2 G^2+3.0 G=13.8 \\

2.0 G^3+F^2=16.6

\end{gathered}

$$

Solve this system of equations using the Newton-Raphson method and starting with $F=G=1.0$ as the initial guess. What is the nature of these equations and do you expect the scheme to converge? Set the system up also as a root-solving problem. Suggest a method to solve it and obtain the solution.

Check back soon!

In a chemical treatment process, the concentrations $c_1, c_2, c_3$, and $c_4$ in four interconnected regions are governed by the system of nonlinear equations

$$

\begin{aligned}

7 c_1+c_2^2+c_3+c_4 & =3.7 \\

c_1^2+8 c_2+3 c_3-c_4 & =4.9 \\

2 c_1-2 c_2+5 c_3+c_4^2 & =8.8 \\

c_1-c_2+\mathrm{c}_3^2+14 c_4 & =18.2

\end{aligned}

$$

Solve these equations by the modified Gauss-Seidel method to obtain the concentrations.

Check back soon!

A manufacturing system consists of a hydraulic arrangement and an extrusion chamber. The two are governed by the following two equations:

$$

\begin{array}{r}

1.3 P-F^{2.1}=0 \\

P^{1.6}-(900-95 F)^{0.5}+10=0

\end{array}

$$

Show the flow of information for this system. Set up the system of equations for an iterative solution, starting with an initial guess of $P=$ $F=1$. Show at least three iterative steps toward the solution.

Check back soon!

Solve the following nonlinear system by Newton's method:

$$

\begin{aligned}

3 X^3+Y^2 & =11 \\

2 X^2+Y-4 & =0

\end{aligned}

$$

Try solving these equations by the successive substitution method as well.

Danielle Leedy

Numerade Educator

In a metal forming process, the force $F$ and the displacement $x$ are governed by

$$

\begin{aligned}

F & =70\left[1-\exp \left\{\frac{1000}{21(5+20 x)}\right\}\right] \\

250 & =4.2 F x

\end{aligned}

$$

Solve for $F$ and $x$, applying the Newton-Raphson method to this system of equations. Also, determine the sensitivity of the force $F$ to a variation in the total input $S, \partial F / \partial S$, where $S$ is the given value of 250 , by slightly varying this input.

Uma Kumari

Numerade Educator

A copper sphere of diameter 5 cm is initially at temperature $200^{\circ} \mathrm{C}$. It cools in air by convection and radiation. The temperature $T$ of the sphere is governed by the energy equation

$$

\rho C V \frac{d T}{d \tau}=-\left[\varepsilon \sigma\left(T^4-T_a^4\right)+h\left(T-T_a\right)\right] A

$$

where $\rho$ is the density of copper, $C$ is its specific heat, $V$ is the volume of the sphere, $\tau$ is the time taken as zero at the start of the cooling process, $\varepsilon$ is the surface emissivity, $\sigma$ is the Stefan-Boltzmann constant, $T_a$ is the ambient temperature, and $h$ is the convective heat transfer coefficient. Compute the temperature variation with time using the RungeKutta method and determine the time needed for the temperature to drop below $100^{\circ} \mathrm{C}$. The following values may be used for the physical variables: $\rho=9000 \mathrm{~kg} / \mathrm{m}^3, C=400 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K}), \varepsilon=0.5, \sigma=5.67 \times 10^{-8}$ $\mathrm{W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}^4\right), T_a=25^{\circ} \mathrm{C}$, and $h=15 \mathrm{~W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}\right)$.

Khoobchandra Agrawal

Numerade Educator

Consider the preceding problem for the negligible radiation case, $\varepsilon=0$, with $h=100 \mathrm{~W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}\right)$. Nondimensionalize this simpler problem and obtain the solution in dimensionless terms. Thus, obtain the results for a 10 -cm-diameter sphere.

Khoobchandra Agrawal

Numerade Educator

The temperature variation in an extended surface, or fin, for the onedimensional approximation, is given by the equation

$$

\frac{d^2 T}{d x^2}-\frac{h P}{k A}\left(T-T_a\right)=0

$$

where $x$ is the distance from the base of the fin, as shown in Figure P4.25. Here $P$ is the perimeter, being $\pi D$ for a cylindrical fin of diameter $D, A$ is the cross-sectional area, being $\pi D^2 / 4$ for a cylindrical fin, $k$ is the material thermal conductivity, $T_a$ is the ambient temperature, and $h$ is the heat transfer coefficient. The boundary conditions are shown in the figure and may be written as

$$

\text { at } x=0: T=T_o \quad \text { and } \quad \text { at } x=L: \frac{d T}{d x}=0

$$

where $L$ is the length of the fin. Simulate this component, which is commonly encountered in thermal systems, for $D=2 \mathrm{~cm}, h=20 \mathrm{~W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}\right)$, $k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, L=25 \mathrm{~cm}, T_o=80^{\circ} \mathrm{C}$, and $T_a=20^{\circ} \mathrm{C}$. Nondimensionalize this problem to obtain the governing dimensionless parameters. Use the shooting method to obtain the temperature distribution, and discuss expected trends at different parameteric values.

(FIGURE CAN'T COPY)

Check back soon!

If radiative heat loss is included in the preceding problem, the governing equation becomes

$$

\frac{d^2 T}{d x^2}-\frac{h P}{k A}\left(T-T_a\right)-\frac{P}{k A} \varepsilon \sigma\left(T^4-T_a^4\right)=0

$$

Solve this problem with $\varepsilon=0.5$ and $\sigma=5.67 \times 10^{-8} \mathrm{~W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}^4\right)$, using the finite difference approach, and compare the results with the preceding problem. Increase the emissivity to 1.0 (black body) in this simulation and compare the results with those at 0.5 . Are the observed trends physically reasonable?

Check back soon!

The temperature distribution in a moving cylindrical rod, shown in Figure P4.27, is given by the energy equation

$$

\frac{d^2 T}{d x^2}-\frac{1}{\alpha} U \frac{d T}{d x}-\frac{2 h}{k R}\left(T-T_a\right)=0

$$

where $U$ is the velocity of the moving rod of radius $R, \alpha$ is the thermal diffusivity, and the other variables are the same as the preceding problem. The boundary conditions are

$$

\text { at } x=0: T=T_o \quad \text { and } \quad \text { at } x=\infty: T=T_a

$$

Employing the finite difference approach, compute $T(x)$. Take $U=$ $1 \mathrm{~mm} / \mathrm{s}, h=20 \mathrm{~W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}\right), \alpha=10^{-4} \mathrm{~m}^2 / \mathrm{s}, k=100 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K}), T_o=$ $600 \mathrm{~K}, T_a=300 \mathrm{~K}$, and $R=2 \mathrm{~cm}$. Numerically simulate $x=\infty$ by taking a large value of $x$ and ensuring that the results are independent of a further increase in this value. Also, nondimensionalize this problem and determine the governing dimensionless variables. If the material and dimensions are fixed, what are the main design variables? Discuss how these may be varied to control the temperature decay over a given distance.

(FIGURE CAN'T COPY)

Check back soon!

For the manufacturing process considered in Problem 3.6, set up the mathematical system for numerical simulation. Take each bolt to have surface area $A$, volume $V$, density $\rho$, and specific heat $C$. Outline a scheme for simulating the process. What outputs do you expect to obtain from such a simulation?

Check back soon!

In the simulation of a thermal system, the temperatures in two subsystems are denoted by $T_1$ and $T_2$ and are given by

$$

\begin{aligned}

& \frac{d T_1}{d \tau}=\left(\frac{h A}{\rho C V}\right)_1\left(T_1-T_a\right) \\

& \frac{d T_2}{d \tau}=\left(\frac{h A}{\rho C V}\right)_2\left(T_2-T_a\right)

\end{aligned}

$$

Under what conditions will the response of $T_1$ be much slower than that of $T_2$ ? Write down the finite difference equations for solving this set of equations and outline the numerical procedure if the time step $\Delta \tau_1$, for $T_1$, is taken as much larger than the time step $\Delta \tau_2$, for $T_2$. Note that $(h A / \rho C V)$ is a function of temperature.

Check back soon!

An experimental study is performed on a plastic screw extruder along with a die to determine the relationship between the mass flow rate $m$ and the pressure difference $P$. The relationship for the die is found to be

$$

m=0.5 \mathrm{P}^{0.5}

$$

and the relationship for the screw extruder is

$$

\mathrm{P}=2+3.5 m^{1.4}-5 m^{2.2}

$$

First, give the block representation for each of these subsystems. Then show the flow of information for the system. Set up this system of equations for an iterative solution by successive substitution, starting with an initial guess of $m=0$ and $P=1$. Obtain the solution to simulate the extruder.

Check back soon!

In an injection molding process, the flow of plastic in two parallel circuits is governed by the algebraic equations

$$

\begin{aligned}

\dot{m} & =\dot{m}_1+m_2 \\

\Delta p & =68+8 \dot{m}^2=550-5 \dot{m}_1-10 \dot{m}_1^{25}=700-10 \dot{m}_2-15 \dot{m}_2^3

\end{aligned}

$$

where $\dot{m}$ is the total mass flow rate, $\dot{m}_1$ and $\dot{m}_2$ are the flow rates in the two circuits, and $\Delta \mathrm{p}$ is the pressure difference. Simulate the system, employing the Newton-Raphson method. Study the effect of varying the zero-flow pressure levels ( 550 and 700 in the preceding equations) by $\pm 10 \%$ on the total flow rate $\dot{m}$.

Check back soon!

Solve the preceding problem using the successive substitution method. The number of equations may be reduced by elimination and substitution to simplify the problem. Compare the results and the convergence characteristics with those for the preceding problem.

Check back soon!

The dimensionless temperature $x$ and heat flux $y$ in a thermal system are governed by the nonlinear equations

$$

\begin{aligned}

& x^3+3 y^2=21 \\

& x^2+2 y=-2

\end{aligned}

$$

Solve this system of equations by the Newton-Raphson and the successive substitution methods, comparing the results and the convergence in the two cases.

Check back soon!

Simulate the ammonia production system discussed in Example 4.6 to determine the change in the ammonia production if the bleed ( 23.5 moles $/ \mathrm{s}$ ) and the entering argon flow ( 0.9 moles $/ \mathrm{s}$ ) are varied by $\pm 25 \%$. What happens if the bleed is turned off? Can this circumstance be numerically simulated?

Lottie Adams

Numerade Educator

The gross production of four substances by an engineering concern is denoted by $a, b, c$, and $d$. A balance between the net output and the production of each quantity leads to the following equations:

$$

\begin{aligned}

4 a-2 b+5 d & =22 \\

-2 a+8 b-c & =16 \\

3 b+4 c-3 d & =30 \\

-3 c+12 d & =6

\end{aligned}

$$

Solve these equations by the SOR method to simulate the system. The constants on the right-hand side of the equations represent the net production of the four items. If the net production of $x_2$ is to be increased from 16 to 24 ( $50 \%$ increase), calculate the gross production of all the items to achieve this.

Check back soon!

A water pumping system consists of pipe connections and pumping stations, each of which has the following characteristics:

$$

P=1850-17.5 \dot{m}-0.7 \dot{m}^2

$$

where $\dot{m}$ is the mass flow rate of water and $P$ is the pressure rise in each pumping station. The mass flow rate is measured as $32 \mathrm{~kg} / \mathrm{s}$ with all eight pumping stations in the pipeline operating. The pressure drop in the pipe is given as proportional to the square of the mass flow rate. Obtain the governing equations to determine the flow rate if a few stations are inoperative and are bypassed. Then calculate the resulting flow rate if one or two stations fail.

Chai Santi

Numerade Educator

The height $H$ of water in a tank of cross-sectional area $A$ is a function of time $\tau$ due to an inflow volume flow rate $q_{\text {in }}$ and an outflow rate $q_{\text {our }}$. The governing differential equation is obtained from a mass balance as

$$

A \frac{d H}{d \tau}=q_{\mathrm{in}}-q_{\mathrm{cut}}

$$

The initial height $H$ at $\tau=0$ is zero. Calculate the height as a function of time, with $A=0.03 \mathrm{~m}^2, q_{\text {in }}=6 \times 10^{-4} \mathrm{~m}^3 / \mathrm{s}$, and $q_{\text {out }}=3 \times 10^{-4} \sqrt{H}$ $\mathrm{m}^3 / \mathrm{s}$. Use both Euler's and Heun's methods with a step size of 10 s . Plot $H$ as a function of time $\tau$. Give the times taken by the height to reach 2 m and 3.5 m in your answer. Why does the increase in $H$ become very slow as time increases?

Khoobchandra Agrawal

Numerade Educator

The temperature of a metal block being heated in an oven is governed by the equation

$$

\frac{d T}{d \tau}=10-0.05 T

$$

Solve this equation by Euler's and Heun's method to get $T$ as a function of time $\tau$. Take the initial temperature as $100^{\circ} \mathrm{C}$ at $\tau=0$.

Audrey Fong

Numerade Educator

A stone is dropped at zero velocity from the top of a building at time $\tau=0$. The differential equation that yields the displacement $x$ from the top of the building is (with $x=0$ at $\tau=0$ )

$$

\frac{d^2 x}{d \tau^2}=g-5 \mathrm{~V}

$$

where $g$ is the magnitude of gravitational acceleration, given as 9.8 $\mathrm{m} / \mathrm{s}^2$, and V is the downward velocity $d x / d \tau$. Using Euler's method, calculate the displacement $x$ and velocity V as functions of time, taking the time step as 0.5 s .

CA

Chi-Chung Ai

Numerade Educator

Simulate the hot water storage system considered in Example 3.5 for a flow rate of $0.01 \mathrm{~m}^3 / \mathrm{s}$, with the heat transfer coefficient $h$ given as 20 $\mathrm{W} /\left(\mathrm{m}^2 \cdot \mathrm{~K}\right)$ and ambient temperature $T_a$ as $25^{\circ} \mathrm{C}$. The inlet temperature of the hot water $T_o$ is $90^{\circ} \mathrm{C}$. Obtain the time-dependent temperature distribution, reaching the steady-state conditions at large time. Study the effect of the flow rate on the temperature distribution by considering flow rates of 0.02 and $0.005 \mathrm{~m}^3 / \mathrm{s}$.

Anand Jangid

Numerade Educator

Consider one-dimensional conduction in a plate that is part of a thermal system. The plate is of thickness 3 cm and is initially at a uniform temperature of $1000^{\circ} \mathrm{C}$. At time $\tau=0$, the temperature at the two surfaces is dropped to $0^{\circ} \mathrm{C}$ and maintained at this value. The thermal diffusivity of the material is $\alpha=5 \times 10^{-6} \mathrm{~m}^2 / \mathrm{s}$. Solve this problem by any finite difference method to obtain the temperature distribution as a function of time.

Check back soon!

A cylindrical rod of length 40 cm is initially at a uniform temperature of $15^{\circ} \mathrm{C}$. Then, at time $\tau=0$, its ends are raised to $100^{\circ} \mathrm{C}$ and held at this value. For one-dimensional conduction in the rod, the temperature distribution $T(x)$ is governed by the equation

$$

\frac{1}{\alpha} \frac{\partial T}{\partial \tau}=\frac{\partial^2 T}{\partial x^2}-H\left(T-T_a\right)

$$

where $H$ is a heat loss parameter. Using any suitable method, solve this problem to obtain the time-dependent temperature distribution for $T_a$ $=15^{\circ} \mathrm{C}, \alpha=10^{-6} \mathrm{~m}^2 / \mathrm{s}$, and $H=100 \mathrm{~m}^{-2}$. Discuss how this simulation may be coupled with the modeling of fluid flow adjacent to the rod and other parts of the system to complete the model of a given system.

Check back soon!

Consider heat conduction in a two-dimensional, rectangular region of length 0.3 m and width 0.1 m . The dimension in the direction normal to this region may be taken as large. The dimensionless temperature is given as 1.0 at one of the longer sides and as 0.0 at the others. Solve the governing Laplace equation by the SOR method and determine the optimum relaxation factor. Discuss how, in actual practice, such a simulation may be linked with those for other parts of the system.

Check back soon!

Consider the fan and duct system given in Example 4.7. Vary the zeroflow pressure, given as 80 in the problem, and the zero-pressure flow rate, given as 15 here, by $\pm 20 \%$. Discuss the results obtained. Are they consistent with the physical nature of the problem as represented by the equations?

Narayan Hari

Numerade Educator

Show the information-flow diagram for Problem 4.18. Also, draw the information-flow diagram for the simulation of Problem 4.35. Do not solve the equations; just explain what approach you will use.

Check back soon!