Good evening people of PF! I have recently encountered a problem from Himmelblau's(adsbygoogle = window.adsbygoogle || []).push({}); Basic Principles and Calculations in Chem. E.which asks to set up an energy balance for a tank, and then non-dimensionalize the differential equation before solving it. It's not the most complex task, but it's the first time I've done it and I wanted to know if I did things right. Here's an sketch of the system.

Mass flows in and out at the same rate w, therefore the mass inside the tank is a constant, m. Heat is being transferred into the contents of the tank using a coiled tube heater, in which steam flows at temperature T_{S}. The rate of heat transfer is given by [itex]\dot{Q} = hA(T_S - T)[/itex]. Fluid entering the tank has a constant temperature of T_{0}, which is also the temperature of the system at t = 0. The differential energy balance for the system is

[tex]\frac{\textrm{d} U}{\textrm{d} t} = \dot{Q} - w \Delta H[/tex]

In terms of temperature

[tex]m C_p \frac{\textrm{d} T}{\textrm{d} t} = hA(T_S - T) - w C_p (T-T_0)[/tex]

The problem only tells you the non-dimensional parameters you are supposed to use

[tex]\Theta = \frac{T-T_0}{T_S - T_0}[/tex]

[tex]\tau = \frac{w}{m} t[/tex]

[tex]\beta = \frac{hA}{w C_p}[/tex]

Which are temperature, time, and an additional constant parameter. In order to start scaling the equation I decided to divide the entire equation by wC_{p}(T_{S}-T_{0}). Also, I've seen some heat transfer textbooks use the relation dT = d(T-T_{0}), because T_{0}is a constant, so I decided to apply said relation and ended up with the following equation

[tex]\frac{m C_p}{w C_p} \frac{\textrm{d}}{\textrm{d} t} \left( \frac{T-T_0}{T_S-T_0} \right) = \frac{hA(T_S - T)}{w C_p (T_S - T_0)} - \frac{w C_p (T-T_0)}{w C_p (T_S - T_0)}[/tex]

[tex]\frac{m}{w} \frac{1}{\frac{m}{w}} \frac{\textrm{d} \Theta}{\textrm{d} \tau} = \beta \frac{T_S - T}{T_S - T_0} - \Theta[/tex]

Using

[tex]\frac{T_S - T}{T_S - T_0} = 1 - \Theta[/tex]

We get the non-dimensionalized energy balance

[tex]\frac{\textrm{d} \Theta}{\textrm{d} \tau} = \beta (1- \Theta) - \Theta[/tex]

[tex]\frac{\textrm{d} \Theta}{\textrm{d} \tau} = \beta - (\beta +1) \Theta[/tex]

In my opinion, it is not necessary to non-dimensionalize the original D.E. in order to solve it, however, I guess it's a good training problem for non-dimensionalization. I was hoping it could get some insights about how to perform the non-dimensionalization procedure more efficiently. Also, I was wondering, if one is not given the non-dimensional parameters to use, is there a set of rules for choosing them or is it just common sense and practice?

Thanks in advance for any input!

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# Non-dimensionalization of the energy balance

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