<Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field>

<Text-field layout="Heading 1" style="Heading 1">Model</Text-field>

Below is a schematic of the heating system. Cold oil is pumped through three identical steam-heated tanks at a constant rate.

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

Standard texts on chemical engineering show that the temperature of the oil in each tank can be modeled by the following system of differential equations:

NiMqJkkiTUc2IiIiIiZJIkNHRiU2I0kicEdGJUYmNiMtSSVkaWZmR0kqcHJvdGVjdGVkR0YlNiQtJkkiVEc2IjYjIiIiNiNJInRHRipGLg== = NiMsJiomSSJXRzYiIiIiLSZJIkNHRiY2I0kicEdGJjYjLCYtJkkiVEdGJjYjIiIhNiNJInRHRiZGJy0mRjE2I0YnRjQhIiJGJ0YnKiZJIlVHRiZGJy1JIkFHRiY2IywmJkYxNiNJInNHRiZGJ0Y2RjlGJ0Yn

NiMqJkkiTUc2IiIiIiZJIkNHRiU2I0kicEdGJUYmNiMtSSVkaWZmR0kqcHJvdGVjdGVkR0YlNiQtJkkiVEc2IjYjIiIjNiNJInRHRipGLg== = NiMsJiomSSJXRzYiIiIiLSZJIkNHRiY2I0kicEdGJjYjLCYtJkkiVEdGJjYjRic2I0kidEdGJkYnLSZGMTYjIiIjRjMhIiJGJ0YnKiZJIlVHRiZGJy1JIkFHRiY2IywmJkYxNiNJInNHRiZGJ0Y1RjlGJ0Yn

NiMqJkkiTUc2IiIiIiZJIkNHRiU2I0kicEdGJUYmNiMtSSVkaWZmR0kqcHJvdGVjdGVkR0YlNiQtJkkiVEc2IjYjIiIkNiNJInRHRipGLg== = NiMsJiomSSJXRzYiIiIiLSZJIkNHRiY2I0kicEdGJjYjLCYtJkkiVEdGJjYjIiIjNiNJInRHRiZGJy0mRjE2IyIiJEY0ISIiRidGJyomSSJVR0YmRictSSJBR0YmNiMsJiZGMTYjSSJzR0YmRidGNkY6RidGJw==

where the parameters are defined as follows:

NiNJIldHNiI= Flow rate of the oil NiMqJkkja2dHNiIiIiJJInNHRiUhIiI=

NiNJIk1HNiI= Mass of oil in each tank NiNJI2tnRzYi

NiMmSSJDRzYiNiNJInBHRiU= Specific heat capacity NiMqJkkiSkc2IiIiIiomSSNrZ0dGJUYmSSJLR0YlRiYhIiI=

NiNJIlVHNiI= Heat transfer coefficient NiMqJkkiV0c2IiIiIiomSSJtR0YlIiIjSSJLR0YlRiYhIiI=

NiNJIkFHNiI= Heat transfer area NiMqJEkibUc2IiIiIw==

NiMmSSJURzYiNiMiIiE= Temperature of cold oil stream NiNJIkNHNiI=

NiMtJkkiVEc2IjYjIiIiNiNJInRHRiY= NiMtJkkiVEc2IjYjIiIjNiNJInRHRiY= NiMtJkkiVEc2IjYjIiIkNiNJInRHRiY= Transient temperatures NiNJIkNHNiI=

NiMmSSJURzYiNiNJInNHRiU= Temperature of steam NiNJIkNHNiI=

We first enter this model into Maple

<Text-field layout="Heading 2" style="Heading 2">Temperature in Tank 1</Text-field>

eq1 := M*Cp * diff(T1(t),t) = W*Cp*(T0-T1(t)) + U*A*(Ts-T1(t));

<Text-field layout="Heading 2" style="Heading 2">Temperature in Tank 2</Text-field>

eq2 := M*Cp * diff(T2(t),t) = W*Cp*(T1(t)-T2(t))+U*A*(Ts-T2(t));

<Text-field layout="Heading 2" style="Heading 2">Temperature in Tank 3</Text-field>

eq3 := M*Cp * diff(T3(t),t) = W*Cp*(T2(t)-T3(t))+U*A*(Ts-T3(t));

<Text-field layout="Heading 2" style="Heading 2">Initial conditions</Text-field>

Initial := T1(0) = T0, T2(0) = T0, T3(0) = T0;

<Text-field layout="Heading 1" style="Heading 1">Deploy</Text-field>

We have used Maple to solve the problem as stated. The final challenge is to put that solution into the hands of the people who can benefit from it. Maple provides numerous options for such deployment. We show two of them below.

<Text-field layout="Heading 2" style="Heading 2">Deployment as a Maplet application</Text-field>

First, we deploy this solution as a Maplet application. It provides a point-and-click interface in which users can experiment with how the system behaves in response to different parameter values. Users don't need to know any Maple commands.or even the theory behind the physical process. If you are viewing this as a live Maple worksheet, click here to run this Maplet application .

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<Text-field layout="Heading 2" style="Heading 2">Deployment to Excel using Visual Basic code generation</Text-field>

Next, we will use Maple's automatic code generation to convert the formulas to Visual Basic code. We can then copy and paste this code into an Excel macro, so that anyone with Excel on their desktop can make use of the solution we have derived.

<Text-field layout="Heading 3" style="Heading 3">Visual Basic code for T1(t)</Text-field>

VisualBasic(simplify(t1), resultname = temp_1, coercetypes= false, optimize);

<Text-field layout="Heading 3" style="Heading 3">Visual Basic code for T2(t)</Text-field>

VisualBasic(simplify(t2), resultname = temp_2, coercetypes= false, optimize);

<Text-field layout="Heading 3" style="Heading 3">Visual Basic code for T3(t)</Text-field>

VisualBasic(simplify(t3), resultname = temp_3, coercetypes= false, optimize);