Entrance Conditions. Chapter 8. Islamic Azad University

Transcription

1 Chapter 8 Convection: Internal Flow Islamic Azad University Karaj Branch Entrance Conditions Must distinguish between entrance and fully developed regions. Hydrodynamic Effects: Assume laminar flow with uniform velocity profile at inlet of a circular tube. Velocity boundary layer develops on surface of tube and thickens with increasing x. Inviscid region of uniform velocity shrinks as boundary layer grows. Does the centerline velocity change with increasing x? If so, how does it change? Subsequent to boundary layer merger at the centerline, the velocity profile becomes parabolic and invariant with x. The flow is then said to be hydrodynamically fully developed. How would the fully developed velocity profile differ for turbulent flow? 2

2 Mean Velocity Velocity inside a tube varies over the cross section. For every differential area da c : Overall rate of mass transfer through a tube with cross section A c : (8.1) (8.2) where u m is the mean (average) velocity! Can determine average velocity at any axial location (along the x- direction), from knowledge of the velocity profile 3 Velocity Profile in a pipe Recall from fluid methanics that for laminar flow of an incompressible, constant property fluid in the fully developed region of a circular tube (pipe): (8.3a) (8.3b) (8.3c) 4

3 Thermal Considerations: Mean Temperature We can write Newton s law of cooling inside a tube, by considering a mean temperature, instead of T! (8.4) where T m is the mean (average) temperature For constant and c p, T m is defined: (8.5) 5 Fully Developed Conditions For internal flows, the temperature, T(r), as well as the mean temperature, T m generally vary in the x-direction, i.e. 6

4 Fully Developed Conditions Although T(r) changes with x, the relative shape of the temperature profile remains the same: Flow is thermally fully developed. A fully developed thermally region is possible, if one of two possible surface conditions exist : Uniform wall temperature (T s =constant) Uniform heat flux (q x =const) Thermal Entry Length : 7 Fully Developed Conditions It can be proven that for fully developed conditions, the local convection coefficient is a constant, independent of x: 8

5 Mean temperature variation along a tube We are still left with the problem of knowing how the mean temperature T m (x), varies as a function of distance, so that we can use it in Newton s law of cooling to estimate convection heat transfer. Recall from Chapter 1, page 10 that by simplifying the energy balance for flow inside a control volume For flow inside a pipe: (8.6) where T m,i and T m,o are the mean temperatures of the inlet and outlet respectively 9 Mean temperature variation along a tube P=surface perimeter For a differential control volume: (8.7) where P=surface perimeter = D for circular tube, =width for flat plate! Integration of this equation will result in an expression for the variation of T m as a function of x. 10

6 Case 1: Constant Heat Flux Integrating equation (8.7): (8.8) where P=surface perimeter = D for circular tube, =width for flat plate 11 Case 2: Constant Surface Temperature,T s =constant From eq.(8.7) with T s -T m = T: Integrating for the entire length of the tube: (8.9) (8.10) where (8.11) A s is the tube surface area, A s =P.L= DL, T lm is the log-mean temperature difference 12

7 Case 3: Uniform External Temperature " Replace T s by and by (the overall heat transfer coefficient, which includes contributions due to convection at the tube inner and outer surfaces, and due to conduction across the tube wall). Equations (8.9) and (8.10) become: (8.11) (8.12) 13 Reminder from Chapter 3, p. 19 lecture notes 14

8 Example (Problem 8.55) Water at a flow rate of kg/s is cooled from 70 C to 30 C by passing it through a thin-walled tube of diameter D=50 mm and maintaining a coolant at 15 C in cross flow over the tube. What is the required tube length if the coolant is air and its velocity is V=20 m/s? The heat transfer coefficients are h i =680 W/m 2.K for flow of water inside the tube and h o =83.5 W/m 2.K for a cylinder in air cross flow of 20 m/s 15 Summary ( ) We discussed fully developed flow conditions for cases involving internal flows, and we defined mean velocities and temperatures We wrote Newton s law of cooling using the mean temperature, instead of (8.4) Based on an overall energy balance, we obtained an alternative expression to calculate convection heat transfer as a function of mean temperatures at inlet and outlet. (8.6) We obtained relations to express the variation of T m with length, for cases involving constant heat flux and constant wall temperature (8.8) (8.9) 16

9 Summary ( ) We used these definitions, to obtain appropriate versions of Newton s law of cooling, for internal flows, for cases involving constant wall temperature and constant surrounding fluid temperature ( ) " In the rest of the chapter we will focus on obtaining values of the heat transfer coefficient h, needed to solve the above equations 17 Heat Transfer Correlations for Internal Flow Knowledge of heat transfer coefficient is needed for calculations shown in previous slides. " Correlations exist for various problems involving internal flow, including laminar and turbulent flow in circular and non-circular tubes and in annular flow. " For laminar flow we can derive h dependence theoretically " For turbulent flow we use empirical correlations " Recall from Chapters 6 and 7 general functional dependence 18

10 Laminar Flow in Circular Tubes 1. Fully Developed Region For cases involving uniform heat flux: (8.13) For cases involving constant surface temperature: (8.14) 19 Laminar Flow in Circular Tubes 2. Entry Region: Velocity and Temperature are functions of x Thermal entry length problem: Assumes the presence of fully developed velocity profile Combined (thermal and velocity) entry length problem: Temperature and velocity profiles develop simultaneously 20

11 Laminar Flow in Circular Tubes For constant surface temperature condition: Thermal Entry Length case or combined entry with Pr"5 (8.15) Combined Entry Length case (8.16) All properties, except s evaluated at average value of mean temperature 21 Turbulent Flow in Circular Tubes For a smooth surface and fully turbulent conditions the Dittus Boelter equation may be used for small to moderate temperature differences T s -T m : (8.17) n=0.4 for heating (T s >T m ) and 0.3 for cooling (T s <T m ) For large property variations, Sieder and Tate equation: (8.18) All properties, except s evaluated at average value of mean temperature 22

12 Turbulent Flow in Circular Tubes The Gnielinski correlation takes into account the friction factor: (8.19) Friction factors may be obtained from the Moody diagram. For small Pr numbers 3x10-3 # Pr #5x10-2 (i.e. liquid metals) (8.20) (8.21) 23 Example (Problem 8.55) Repeat Problem This time the values of the heat transfer coefficients are not provided, therefore we need to estimate them. Water at a flow rate of kg/s is cooled from 70 C to 30 C by passing it through a thin-walled tube of diameter D=50 mm and maintaining a coolant at 15 C in cross flow over the tube. (a) What is the required tube length if the coolant is air and its velocity is V=20 m/s? (b) What is the required tube length if the coolant is water is V=2 m/s? 24

13 Non-Circular tubes Use the concept of the hydraulic diameter: where A c is the flow cross-sectional area and P the wetted perimeter! See Table 8.1 textbook for typical values of Nusselt numbers for various cross sections 25 Example (Problem 8.80) You have been asked to perform a feasibility study on the design of a blood warmer to be used during the transfusion of blood to a patient. It is desirable to heat blood taken from the bank at 15 C to a physiological temperature of 37 C, at a flow rate of 200 ml/min. The blood passes through a rectangular cross-section tube, 6.4 mm by 1.6 mm, which is sandwiched between two plates held at a constant temperature of 40 C. Compute the length of the tubing required to achieve the desired outlet conditions at the specified flow rate. Assume the flow is fully developed and the blood has the same properties as water. 26

14 27 Summary Numerous correlations exist for the estimation of the heat transfer coefficient, for various flow situations involving laminar and turbulent flow. Always make sure that conditions for which correlations are valid are applicable to your problem.! Summary of correlations in Table 8.4 of textbook 28