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tank of liquid [WAS: Birthday Wish]



Hello all:

Brian Whatcott <inet@intellisys.net> wrote:

At 14:39 4/12/99 +1000, Brian McInnes wrote:
Brian,
You pose a question

I want to bring up a tank of liquid from 125degF to 140degF using
hot liquid at 190degF. I want to raise its temperature at 4 degrees
a minute. What flow rate should I use?

(This question is ill-posed enough to be 'real-world', don't you
think?
Could it be a dairy-farmer has an application for this answer?
Just for fun, try giving an exact formulation assuming no losses.

An attack on this "real-life" situation requires
understanding of a number of concepts that play a part in a
physicist's model of thermal physics: thermal energy,
thermal contact, thermal equilibrium, temperature, energy
transfer, rate of energy transfer.
A naive application of a derivative of the formulation you
called attention to in your previous posting
mass1 x specific heat1 x rise in temperature =
mass2 x specific heat2 x fall in temperature
does not take into account many of the consequences of these
concepts in the application of the model.
For example, the amount of thermal mixing, the mechanics of
the thermal contact between the hot liquid and the liquid in
the tank, could well be more important in the warming of the
tank liquid than the flow rate of the hot liquid.

Brian McInnes

Please assume perfect, fast mixing as well as no losses.
That should provide sufficient input for either a naive,
an approximate, even an exact flow rate determination,
as you prefer.

Assuming perfect, fast mixing, no losses due to radiation/conduction,
and identical fluids (same density and heat capacity), the following
formula describes the temperature of the 'milk' as a function of time:

V0 * T0 + V(t) * T1
T(t) = ---------------------
V0 + V(t)

Where:

t is the time
T(t) is the temperature of the resulting milk
V0 is the initial volume of the milk
T0 is the initial temperature of the milk
V(t) is the volume of milk that has been added at time 't'
T1 is the temperature of the added milk

Proof of the formula is left as an exercise for the student.

Solving this equation for a flow rate (dV/dt) that yields a constant
rate of temperature change ('4 degrees a minute') looks really tough.

Regards,
John