Flows: Fluids, Heat, Electricity

(About flow of fluid, heat energy, and electrical charge: all modeled after Ohm's law, the electrical case) Do not try to apply this method to the wind, fires, lightning, movement of glaciers, flow of material in Earth's mantle…. These are too complex for a simple model.

            Fluid flow rate through a pipe or duct is proportional to the pressure difference
(D P) from one end to the other. The fluid then flows from the higher to the lower P. (Important note:
If there is a difference in elevation D h between the ends, the lower end will be
r gD h higher in pressure due to gravity, with no flow (r = density = mass/volume). So the D P that
we will use is the difference, if any, between the actual
D P and the D P due to gravity.)

            Heat energy conduction through a slab of material is proportional to the temperature difference (D T) between the ends. The heat (energy) flows from the higher to the lower T.

            Electric charge flow through a material is proportional to the voltage difference
(D V) between the ends. The flow (current) is from the higher to the lower V. (This is a convention; if electrons are flowing, they go the other way. To maintain mathematical consistency, however, we define current direction to be opposite to the direction of electron flow, because the electron is negative.)

            In all three cases there is a resistance to flow (R). In the case of fluids, it is due to the gooeyness of it (viscosity is the tech term), and the length and cross-sectional area of the pipe. In heat flow, high R means that the vibrating molecules do not readily pass along their energy of vibration to their neighbors. In all of these flows, a larger cross-sectional area means a smaller R and greater length means larger R.

The equations:

fluids- dQ/dt = D P/R, where dQ/dt is volume transfer/time

 

heat- dQ/dt = D T/R, where dQ/dt is heat energy transfer/time

(but see note below on heat)

 

elec.- dQ/dt = D V/R, where dQ/dt is charge transfer/time (=current)

 

SI units:            volume, m3; pressure, N/m2; so Rfluids is in Ns/m5

                        energy, J; temperature K, so Rheat is in Ks/J

                        charge Coulomb (C); voltage V; so Relec is in Vs/C

 

Odds & ends:

Fluids- a measure of the gooeyness of a fluid is viscosity, h . It can be shown (beyond this course) that for a straight pipe of radius r and length L, R = 8h L/(p r4).

 

Heat-The convention for insulation is to define an "R value," Â . This is defined to be independent of cross sectional area. In terms of this, the resistance to flow defined above is R = Â /A, and the units are such that A is in ft2, D T is in oF and dQ/dt is in BTU/hr. Fiberglass insulation six inches thick is labeled R-19 in the USA. This means Â=19 hr oF ft2/BTU.

 

Electricity- A coulomb of charge flow per second is known as an Ampere (or Amp), and everyone uses the symbol I or i for this. Also, the D is omitted. So I = V/R or V = IR is the standard way of writing the Ohm's law equation.

Series and parallel:
Rs = S R and Rp = 1/S (1/R) can be found in any physics book in the section on electric current or go here, but it is worth noting that these apply to heat and fluid flow, as well. In the winter, heat is flowing out of your house by different parallel paths.

flow back to the main page on fluids, etc. Other main pages:

mechanics
vibrations and waves
quantum

Or look up stuff in my index, or bug me by email: fredrick.gram@tri-c.edu