Flow in an Annulus with changing κ

Submitted by Kartik Jain (00002014) Undergraduate student Department of Chemical Engineering IIT-Bombay.

Acknowledgement

I would like to thank Prof. G.K. Suraishkumar for introducing me to the laboratory which had a creative way of working and also inspiring us throughout the lab for being innovative and practical.

Flow in an Annulus with changing κ

Objectives

1. To investigate experimentally the relation between volumetric flow rate and κ (ratio of the internal to external radius of the annulus) by changing the internal diameter.

2. At a particular κ, observe the laminar and turbulent flow in an annulus and determine the critical Reynolds number for transition from laminar to turbulent.

3. For the κ in part 2, relate the friction factor and the Reynolds number for flow in annulus.

Theory The volumetric flow rate is related to κ and the pressure drop by the following equation:

Q = π P R4 ((1- κ 4) – (1- κ 2)2 ) ..……..………… (1) [1] 8 µ L ln(1/ κ)

where, P = pressure drop across the annulus. κ = ratio of internal to external radius of the annulus. L = length of the annulus. R = outer radius of the annulus. The friction factor is given by: f = D h P ………………… (2) [3] 2L ρv2 where D h (Hydraulic diameter) = 4 * Cross sectional area [2] Wetted Perimeter v = velocity of the fluid. ρ = density of the fluid. Reynolds number for an annulus is given by: Re = 2R (1- κ) v ρ ………………...(3) [1] µ

Experimental setup The apparatus consists of a transparent glass tube (which will comprise the external diameter) and a rubber tube (which will comprise the inner diameter), which can expand freely. To ensure that the rubber tube does not slack in between there is a thin rod inside the rubber tube (coaxial with the outer glass tube) which will support the tube and maintain its stability, i.e. keep the tube horizontal. The rubber tube is closed at one end and the other end is connected to a compressor. We can inflate the tube uniformly (as the tube of a bicycle expands uniformly) throughout the length to get the internal diameter. To avoid the end effects of expansion enough margin can be left at the end (especially at the fixed end, as some distortion is expected). In case we need to deflate the tube, there will be a valve provided (similar to tubes used in bicycles). To measure the internal diameter of the annulus i.e. the diameter of the rubber tube we will use a laser light which will be movable on a linear scale for accurate measurement of internal diameter. For the 2nd part of the experiment (Reynolds number), the level of the nozzle can be adjusted so as to get it at the middle of the outer and inner diameters and ensure that the dye travels in the center to get a clear view of laminar flow as well as transition. A manometer is connected across the length of the pipe to get the pressure drop across the annulus. The flow rate will be measured manually with a measuring cylinder and stop watch.

Front View

Outer radius

Thin rod

Inner radius (κR, made of rubber)

Valve connected to compressor for inflation nozzle

Manometer (for pressure measurement)

Tank A Tank B

Dye

Procedure Varying Inner Diameter of Annulus

1. Adjust the inner diameter of the tube to a minimum by adjusting the flow of air through the compressor.

2. Allow water to flow through the annulus at a fixed pressure difference (say ∆h in the manometer tube = 25 cm), by adjusting the flow rate of water.

3. Measure the flow rate of water manually, using a measuring cylinder and a stop watch.

4. Measure the internal diameter of the annulus using the laser beam by moving it on a linear scale.

5. Change the inner diameter of the tube by allowing more air from the compressor

and adjust the flow rate so as to get the same pressure difference as above and note down the readings again.

6. Get the Q vs. κ relation from the experimental results and compare it with the theoretical results from equation 1.

Laser light

Graph paper attached to get the reading of the diameter

Stand for the support of the laser light

Tank B (side view) Inner Diameter

Reynolds Number experiment

1. For a particular value of κ (keeping it constant) adjust the nozzle of the dye by bringing it to the middle of flow and set the flow rate of the dye to a constant value.

2. Observe the flow pattern inside the annulus at different flow rates. 3. Note the flow rates and the corresponding pressure difference across the length. 4. Calculate the Re and the friction factor f for each flow rate using eqns. 2 and 3. 5. Compare the plot of f vs. Re with the theoretical results.

Critical Reynolds Number For calculating the critical Reynolds number accurately, first calculate Re with increasing flow rate and then with the decreasing flow rate. References:

[1]. Bird,R.B.;Stewart,W.E.;Lightfoot,E.M. Transport Phenomena. pp 51-54. [2]. McCabe,W.L.;Smith,J.C.;Harriot,P.Unit Operations in Chemical Engineering. 5th Edition. pp 103 [3]. http://www.processassociates.com/process/dimen/dn_dar.htm