First Step in Modeling. An iron sphere with radius of 2 cm at initial

temperature of 100 C is immersed in a 53 water container with temperature of 20 C. If temperature distribution inside the sphere is lumped. Gain

temperature profile of the sphere and the water inside

the container after 10 minutes. Consider the water

volume constant and exert conduction heat transfer

coefficient considering the surrounding water at rest.

Solution.

The first step is to gather data completely and represent the problem effectively.

The second step is to write the governing equations of the phenomenon. Since equations gained by

energy balance have the temperature variable, solving them would generate a temperature profile.

Hence we start by writing energy balance for the two control volumes.

{1: = ; i

2: + =

1( ) =

; The terms must be dimensionally balanced.

( ) =

( 103); (:

=42

43

3=3

)

( ) = 10

3

3

103....( ) =

( ) = [1]

Eq.1 is the governing equation of the control volume 1, solving this equation will result

in a function of temperature (for the sphere/water) with regard to time. But as can be

seen the equation has two dependent variables and time as one independent variable. It

is an ODE which its dependent variables also depend on each other. In case was constant this ODE would be easily solved by defining a parameter (i.e. = ), however is not constant. We need another equation with two variables , . Hence we write the energy balance for the second control volume. 2 + ( ) =

103.

+ = + + ( ) =

C

1

( + + ) =

[2]

Using eqs.1, 2 we have an ODE system of equations:

{

( ) =

1

( + + ) =

In order to solve this system we need two initial values: at t=0 { = 100 = 20

,

ode45 in Matlab will solve these systems. Hence the result is:

Matlab script

function []=sphere()

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% This script generates the results for hot sphere problem. %

% 12/2013 by Arash Nasiri. %

% arash.nasiris@gmail.com %

% This script is published on line for educational purposes so feel free to modify and use. %

% %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Tsio=100;

Twio=20;

x0=[Tsio Twio];

tspan=[0,600];

[t,x]=ode45(@sub,tspan,x0)

plot(t,x(:,1),'r',t,x(:,2),'b',[0:30:600],25,'.g')

xlabel('Time[s]')

ylabel('Temperature[C]')

axis([0,600,15,100])

legend('Sphere Temp','Water bath Temp')

end

function xprime = sub(t,x)

rs=.02; %[m] radius of the immersed sphere

Vt=5; %[cubic m] total volume of the bath(which is constant!)

As=4*pi*(rs^2); %[m^2] interference cross section(between sphere and the bath)

Cs=8.32/1000;

ROs=7830; %[kg/m^3] density of the sphere

Ms=55.84; %[g-mol/mol] molecular weight of the sphere's substance (here iron)

Mw=18.02; %[g-mol/mol] molecular weight of the bath's substance(water)

ROw=1000; %[kg/m^3] density of the bath substance(water)

Tin=25; %[C] the constant temperature of the entering fresh flow into the bath

Cw=4.18*18.02; %[J/mol.C] heat capacity of the bath substance(water)

q=0.01; %[m^3/sec] debi of the entering flow

m=q*Row*Mw*1000; %[mol/sec] debi of the entering flow

h=400; %[W/m^2.C] convection's heat transfer coefficient for water

%*************************************

Alpha=-(h*3)/(1000*Ms*ROs*Cs*rs);

Beta=(1000*Mw*ROw*Cw*Vt);

C=m*Cw*Tin;

Del=h*As;

Gamma=-h*As-m*Cw;

%*************************************

xprime=[Alpha*x(1)-Alpha*x(2);(1/Beta)*(Del*x(1)+Gamma*x(2)+C)];

end

i A lumped system is one in which the dependent variables of interest are a function of time alone. In general, this will mean

solving a set of ordinary differential equations (ODEs), meanwhile a distributed system is one in which all dependent

variables are functions of time and one or more spatial variables. In this case, we will be solving partial differential

equations (PDEs)