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 . Internal Flow 20

Laminar Flow in Circular Tubes Fully Developed Region We start from the energy equation, written for fully developed, flow in one direction and substitute known velocity profile for flow in tubes where u=0 and For constant heat flux, the solution of the differential equation is: Combining with Newton’s law of cooling: Internal Flow 21

Laminar Flow in Circular Tubes For cases involving uniform heat flux: (8.17) For cases involving constant surface temperature: (8.18) Internal Flow

Laminar Flow in Circular Tubes 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 Internal Flow 23

Laminar Flow in Circular Tubes For constant surface temperature condition: Thermal Entry Length case (8.19) Combined Entry Length case (8.20) All properties, except ms evaluated at average value of mean temperature Internal Flow 24

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 Ts-Tm: n=0.4 for heating (Ts>Tm) and 0.3 for cooling (Ts<Tm) (8.21a) For large property variations, Sieder and Tate equation: (8.21b) All properties, except ms evaluated at average value of mean temperature Internal Flow 25

Turbulent Flow in Circular Tubes The effects of wall roughness may be considered by using the Petukhov correlation: (8.22a) For smaller Reynolds numbers, Gnielinski correlation: (8.22b) Friction factors may be obtained from Moody diagram etc. Internal Flow 26

Example (Problem 8.57) Repeat Problem 8.57. This time the values of the heat transfer coefficients are not provided, therefore we need to estimate them. Water at a flow rate of 0.215 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? What is the required tube length if the coolant is water is V=2 m/s? Internal Flow 27

Non-Circular tubes Use the concept of the hydraulic diameter: where Ac 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 Internal Flow 28

Example (Problem 8.82) 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 10°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. Internal Flow

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 Internal Flow 29