LECTURE 1 THE CONTENTS OF THIS LECTURE ARE AS FOLLOWS: 1.0 INTRODUCTION 2.0 ATKINSONS EQUATION 3.0 DETERMINATION OF COEFFICIENT OF FRICTION FACTOR (k) 3.1 By Analogy With Similar Airways 3.2 From Design Tables 3.3 From Geometric Data REFERENCES Page 1 of 11
1.0 INTRODUCTION We have already learnt about the head loss in case of laminar flow as well as turbulent flow. We have also concluded that laminar flow rarely exits in mines. All of us know that energy is required for flow of air in mines. Head loss of travelling air in mines suggests that a resistance to airflow exists during the course of its travel. Head loss derived in case of mines is dependent on factors like roughness of walls, velocity of air, shape and size of airways, etc. Atkinson was the first one who came up with qualitative approach of RESISTANCE OF MINE AIRWAYS and parameters governing it. Since, then ATKINSON S EQUATION is one of the most widely accepted and used equation for underground mine environment. Let us see expression for Atkinson s equation. 2.0 ATKINSON S EQUATION From Chezy-Darcy equation we have p = λρlu2 2d We know that mean hydraulic diameter, d is given by d = 4A per Substituting the value of d = 4A per in Chezy-Darcy equation, we have p = λρl per u2 8A Where, Page 2 of 11
p = frictional pressure drop (Pa) λ= Darcy Weisbach resistance coefficient or resistance coefficient (dimensionless) ρ= Density of air (Kg/m 3 ) L = Length of the airway (m) per = Perimeter of the airway (m) u= Velocity of air (m/s) A = Area of cross section of the airway (m 2 ) The above equation can be written in the form p = klper u2 A, where k = λρ 8 We know that air flow i.e. quantity of air-flow is related with velocity of air and cross-section area of airways as u = q A Substituting for u = q A in the above equation we have p = k L per q2 A 3 = R q 2 Where, R = k L per A 3 R is known as Atkinson s resistance or frictional resistance of airway which has units of Ns 2 /m 8 Page 3 of 11
q= Air flow (m 3 /s) k = Atkinson s friction factor or coefficient of Atkinson s resistance (Ns 2 /m 4 ) From the expression for Atkinson s resistance, we can easily mention the parameters on which the frictional resistance of a mine airway depends. They are: a. Length of the mine airway (L) R L b. Cross section area of mine airway (A) R 1 A 3 c. Perimeter of mine airway opening (per) R per Note : The parameters length and perimeter together constitute the rubbing surface area (s = L per). Hence, R s. Having discussed all the three parameters related to shape and size we can also say that d. Roughness of wall R λρ 8 = k Where λ = Coefficient of friction (describing roughness of wall) Page 4 of 11
If we keep the density of air constant throughout the course of its travel in a mine, k attains a constant value. But, ideally it is not possible, especially when the length of airway is large. Hence, we apply a density correction to Atkinson s equation when we make use of a constant value of k (calculated for air density 1.2) for calculation. The modified Atkinson s equation is given by p = k L per q2 A 3 = R q 2 (for air-density 1.2) p = k L per q2 A 3 ρ 1.2 = R q2 ρ 1.2 (for variable air density) k, in the above two equations is constant (same value for both equation) and is given by, k = 1.2λ 8 Let me tell you that Atkinson s equation is also called SQAURE LAW because the pressure drop is proportional to the square of the volume of air-flow rate. You must we wondering, why do we apply density correction formula and constant value of k for calculation. Instead we can use a variable value of k according to density variation of air. Both are same thing. But the former one is easy to apply. The reason being, we need to determine the value of k each time which is a very cumbersome task compared to using density correction formula for better results. Page 5 of 11
Fig.1 Graph Illustrating Square Law 3.0 DETERMINATION OF COEFFICIENT OF FRICTION FACTOR (k) First of all question arises that why do we need to determine the coefficient of friction factor,k. We all know that expansion of mining areas or development is very common in mining practices. In underground mining, development includes making galleries, drifts and panels. Many times, we may seal some part of mining areas. All these aspects are to be taken into account while planning of a mine. At this stage, we have to make an approximation of the resistance that would be encountered and then the subsequent cost of ventilation, the power of fans to be setup, etc. Hence, we Page 6 of 11
estimate resistance as well as cost of ventilating mine galleries or airways even when their construction has not started. Here, we also need to plan the shortest possible mine airway network with highest effectiveness. Thus, determination of coefficient of friction factor is a must. Various methods are there for determining coefficient of friction factor. The three main methods generally employed are (after McPherson, 1993): o By analogy with similar airways o From design table o From geometric table 3.1 By Analogy with Similar Airways In this method of determining k we choose some airways as representatives or standard airways. These representatives or standard airways may be from intakes, returns, conveyor roadways, roadways consisting of a particular support system. For these airways, their geometry and density of air in them are determined. Then the corresponding value of friction factor is calculated for these airways and referred to standard density as: K1.2 = p Q 2 A3 L per 1.2 ρ kg/m3 These values of friction factor are then standardized and are used to calculate the resistances of other similar airways for different air densities also. The factor 1.2 / is called density correction and we can use it to calculate k values for airways of different densities. Page 7 of 11
While using these standard values one should be careful that any unwanted blockages or obstructions in those airways are not overlooked. This method is advantageous in cases where the number of airways is too many and vast amount of data is to be collected. 3.2 From Design Tables According to this method, the value of friction factor is calculated based on the empirical data of ventilation surveys which have been obtained through various tests and observations in various mines, countries and airway conditions. Such tables should be used only as a guide and only when the friction factors pertaining to a particular area is not available. Table 1 gives the friction factor at air density of 1.2 kg/m 3 and the coefficient of friction (independent of air density). Page 8 of 11
Table 1 Friction factors and coefficients of friction (after, McPherson, 1993) Rectangular airways Friction factor, k (kg/m 3 ) Coefficient of friction, f* (dimensionless) Smooth concrete lined 0.004 0.0067 Shotcrete 0.0055 0.0092 Unlined with minor irregularities only 0.009 0.015 Girders on masonry or concrete walls 0.0095 0.0158 Unlined, typical conditions, no major irregularities 0.012 0.020 Unlined, irregular sides 0.014 0.023 Unlined, rough or irregular conditions 0.016 0.027 Girders on side props 0.019 0.032 Drift with rough sides, stepped floor, handrails 0.04 0.067 Steel arched airways Smooth concrete all round 0.004 0.0067 Bricked between arches all round 0.006 0.010 Concrete slabs or timber lagging between 0.0075 0.0125 flanges all round Slabs or timber lagging between flanges to spring 0.009 0.015 Lagged behind arches 0.012 0.020 Arches poorly aligned, rough conditions 0.016 0.027 Shafts Smooth lined, unobstructed 0.003 0.005 Brick lined, unobstructed 0.004 0.0067 Concrete lined, rope guides, pipe fittings 0.0065 0.0108 Brick lined, rope guides, pipe fittings 0.0075 0.0125 Unlined, well trimmed surface 0.01 0.0167 Unlined, major irregularities removed 0.012 0.020 Unlined, mesh bolted 0.014 0.023 Tubbing lined, no fittings 0.007 0.014 0.0012 0.023 Brick lined, two side bun tons 0.018 0.030 Two side buntons, each with a tie girder 0.022 0.037 Longwall faceline with steel conveyor and powered supports Good conditions, smooth wall 0.035 0.058 Typical conditions, coal on conveyor 0.05 0.083 Rough conditions, uneven faceline 0.065 0.108 Ventilation ducting Collapsible fabric ducting (forcing systems only) 0.0037 0.0062 Flexible ducting with fully stretched spiral spring reinforcement 0.011 0.018 Fibreglass 0.0024 0.0040 Spiral-wound galvanized steel 0.0021 0.0035 *Average values of f (=λ/4) and k for air density=1.2 kg/m 3 Page 9 of 11
3.3 From Geometric Data In this method, the friction factor is calculated as a function of the ratio e/d, where e = height of the roughening d= hydraulic mean diameter of the airway or duct (d = 4A/per) The application of this method is limited to those areas where the height of the roughenings can be measured. The technique is applicable for supports that project a known distance into the airway. The curve given in Fig. 2 is according to Von Karman equation, which is -: k1.2 1 λ/4 = f 2 0.6 4[2 log (d/e) 1.14] 10 Fig. 2 Variation of coefficient of friction with the height of roughening to airway diameter ration. e/d (after, McPherson, 1993) Page 10 of 11
REFERENCES Banerjee S.P. (2003); Mine Ventilation ; Lovely Prakashan, Dhanbad, India. Hartman, H. L., Mutmansky, J. M. & Wang, Y. J. (1982); Mine Ventilation and Air Conditioning ; John Wiley & Sons, New York. McPherson, M. J. (1993); Subsurface Ventilation and Environmental Engineering ; Chapman & Hall, London. Misra G.B. Calcutta, India. (1986); Mine Environment and Ventilation ; Oxford University Press, Vutukuri, V. S. & Lama, R. D. (1986); Environmental Engineering in Mines ; Cambridge University Press, Cambridge. Page 11 of 11