Convection In Internal Flows

March 17, 2004 • Course Evaluations Monday • Fuel Cell Centre in Kingston (1.7 Million Funding so Far) Convection in internal Flows contd. Continuing on from last day, we can easily solve the energy equation in cylindrical coordinates, using the parabolic velocity profile shown last day for laminar flow in a pipe, to determine

For constant flux boundary conditions and laminar fully developed flow, and

for constant surface temperature conditions and laminar fully developed flow. It is not at all surprising that the heat transfer is lower in the case of constant surface temperature since in this case the difference between the surface temperature and the mean temperature decreases with distance along the pipe, and hence the heat transfer decreases. Entry Region

Heat transfer rates are higher initially due to the higher gradients in the flow, as anticipated from the boundary layer solutions in the previous chapter. The heat transfer rates will decrease with distance along the pipe, and approach asymptotically the fully developed values above. Fully Developed Turbulent Flow It is far more difficult to develop an analytic solution for turbulent flow, do we rely on empirical correlations. One example of an empirical correlation is

Be careful of the details and range of applicability when using this equation, as with all correlations. Be especially careful about the effect of property variations with temperature. This correlation is valid for both constant flux conditions, and for constant temperature conditions. Non-Circular Replace D, the pipe diameter, with a new length scale, the hydraulic diameter

• For turbulent flow, use Dh, and proceed as normal • For Laminar flow, use table 8.1 in text

Example Problem Consider a thin walled metallic tube of length L=1m and Diameter, D=3mm. Water enters the tube at a mass flow rate mdot = 0.015 kg/s and Tm,i =97oC a) What is the outlet T of the water if the tube surface is maintained at 27oC? b) If a 0.5mm thick layer of insulations of k=0.05 W/mK is applied to the tube and its outer surface is maintained at 27oC, what is the outlet T of the water? c) If the outer surface of the insulation insulation is exposed to air at 27oC with a free convection coefficient of 5 W/m2K, what is the outlet T of the water? Solution Assumptions • Steady state conditions • Fully developed conditions (hydraulic and thermal) • Negligible conduction resistance from the metal pipe wall. • Uniform convection at the outer surface

a) Material Properties We must assume an outlet temperature for the purpose of determining fluid properties at the average of the inlet and outlet temperatures. We shall assume an oulet T of 27oC. The average T is then (97 + 27)/2 = 62oC = 335K Table A-6 for water, Cp = 4186 J/KgK, mu = 453e-6 Ns/m2, k= 0.656 W/mK and Pr = 2.88 First lets calculate the Reynolds number to determine if we have laminar or turbulent flow

Therefor the flow is turbulent

Now that we know the convection coefficient we can use the energy balances from last day.

We assumed this was 27oC Æ this is close enough for a lecture, or for an exam. b) When there is insulation, we need to replace h with an overall heat transfer coefficient, U which includes conduction and convection Because of the insulation however, there will be less heat transfer to the water, and we can expect a higher outlet temperature. This means that the material properties will change. Assume negligible heat transfer Æ Tm,o = 97oC. Evaluate properties at 97oC = 370K Table A-6 cp = 4214 J/kgK mu = 289e-6 NS/m2, k = 0.679 W/mK and Pr = 1.8

Therefor the flow is still turbulent. Note that the Reynolds number has changed significantly due to the property variation with T.

Now proceed exactly as before, using conservation of energy

Our assumption of the outlet temperature for the property calculation was well justified. c) In this case we are simply adding one resistor to our network to account for convection from the outer surface of the insulation. We can again assume

negligible heat transfer (there is more resistance now) and thus evaluate the properties at 370K as in b). Re is this exactly as it was in b) as well. Our new overall heat transfer coefficient including the third resistor is

And we can proceed exactly as before.