Jökull


Jökull - 01.12.1968, Side 27

Jökull - 01.12.1968, Side 27
where z0 is the equivalent sand roughness of the bottom and x is von Kármán’s constant (x ~ 0.4). vf is the friction velocity defined by Vf = V—= V gOI (6) v e where x0 is the shear stress at the bottom, o the specific mass (o = Y/g). g the acceleration of gravity, D the depth and I is the slope of the energy line. The equation (5) is not ac- curate close to the surface and it does not hold in the laminar layer at the bottom. The diffusion coefficient KTz is of the same order of magnitude as the corresponding coeffi- cient for momentum transfer, KJf (the eddy viscosity), which is defined by: v r —________________K ’ Y •' V. ~~ 3vx 3z“ the transfer of heat being similar to transfer of momentum (Reynolds analogy). An expres- sion for Km in our case is obtained as follows: Km = VX T/e dv/dz dv/dz x is the shear stress at height z: z T = T„ 1 — and from (5) With this we get D dv : Vf2 6 1 dz xz D Km = «vfz f D (7) The ratio KM/KT is the dimensionless turbu- lent Prandtl number, (8) In turbulent flow the value of Pt is close to unity at a boundary wall but decreases to ap- proximately 0.5 far away from it (Schlichting 1960, p. 499). A comparison between k/yc and KTz can now be made for a practical case, say a river with D = 2.0 m and I = 0.002. It is assumed that Pt = 0.8. For this case the following values of KTz are found: z/D 0.95 0.75 0.50 KTz [m2 s-1] 0.0094 0.0372 0.0496 k/yc is a function of the water temperature: T [°C] 0 10 k/yc [m2 s-1] 1.3 - 10-7 1.4 • 10-v It is seen that in this case the heat exchange by turbulent diffusion is about 100 000 times greater than by conduction, except very close to the surface and the bottom where KTz^> 0. KTx can not be calculated as KTz. It must be a function of z and probably ol the same order of magnitude as KTz. For sufficiently high flow velocities we have then 3 3x 3T 3T ---«v---- 3x 3x so that in many cases diffusion in the flow direction can be neglected. Doing this and in- serting the above expressions for v and KTz we get a differential equation for the water temp- erature in turbulent flow in a wide rectangular channel: 3T "aT ■+ Vf 3 3z — ln — + 8.48) — x z„ / 3x k --+ ■ xv,z 1 3T 3z YC »t The boundary condition at the surface is 3T (9) 'dz — — S; (10) where S is the heat loss froin surface per unit time and unit area. The boundary condition at the bottom is similar and in most practical cases we can neglect the heat exchange with the bottom. To find the dimensionless groups on which the solution to (9) must depend we introduce dimensionless quantities, denoted by star: JÖKULL 18. ÁR 361

x

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