1/22/05ME 2591 ME 259 Heat Transfer Lecture Slides IV Dr. Gregory A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico

1/22/05ME 2592 Natural (Free) Convection Heat Transfer

1/22/05ME 2593 Natural Convection Fundamentals u Heat transfer is due to fluid motion by buoyancy forces u Buoyancy arises from a density variation in the fluid, most commonly due to a temperature gradient: u Velocities are generally much smaller than those associated with forced convection, hence –For air, h = 2 – 10 W/m 2 -K –For H 2 O, h = 50 – 1000 W/m 2 -K

1/22/05ME 2594 Natural Convection Fundamentals, cont. u Radiation and natural convection heat transfer are comparable in most applications u Analysis often leads to an iterative solution technique ( IHT ! ) since q rad = f (T 4 ) and q conv = f (T 4/3 ) or f (T 5/4 ) u Natural convection strongly influences –Heating & cooling of rooms –Cooling of electronics, engines, refrigeration coils, transmission lines –Human and animal comfort –Environmental pollution –Atmospheric motions –Oceanic currents

1/22/05ME 2595 Natural Convection Fundamentals, cont. u Physical considerations – consider a quiescent fluid between parallel, heated horizontal plates:

1/22/05ME 2596 Physics of Natural Convection u Consider a wood stove or fireplace: Buoyant force per unit area:

1/22/05ME 2597 Physics of Natural Convection, cont. u Now consider a vertical heated plate: Buoyant force per unit area: u The quantity (   -  ) is related to a thermodynamic property called the volumetric thermal expansion coefficient (  ):

1/22/05ME 2598 Physics of Natural Convection, cont. u This buoyant force will cause fluid motion that is resisted by viscous forces: u The ratio is important in determining the magnitude of natural convection - known as the Grashof number (Gr):

1/22/05ME 2599 Important Dimensionless Parameters in Natural Convection u For a particular geometric shape and orientation, u Dimensional analysis shows that u The Grashof number plays a similar role to the Reynolds number in forced convection u The product Gr Pr appears frequently in analysis of natural convection, so we also define the Rayleigh number ( Ra ):

1/22/05ME Empirical Correlations for Natural Convection Heat Transfer u Many natural convection geometries have been studied; the correlations are typically written as u Geometries: –vertical plate, vertical cylinder –horizontal plate –horizontal cylinder –sphere –array of horizontal cylinders –inclined plate –parallel plates, or channel –rectangular cavity, or enclosure –annular cavity

1/22/05ME Thermal Expansion Coefficient,   For an ideal gas,  = P/RT, so  For liquids and non-ideal gases,  is found from Appendix tables in text, e.g., A.4 - A.6

1/22/05ME Combined Natural and Forced Convection u Both types of convection are comparable when –the resulting flow field is complicated and difficult to predict; it is strongly influenced by the direction of the buoyancy relative to that of the flow; the effect on Nu L can be estimated for three special cases by –where n = 3for vertical plates & cylinders n = 3.5for horizontal plates n = 4for horizontal cylinders & spheres and+ refers to assisting and transverse flows - refers to opposing flows

1/22/05ME Heat Exchangers

1/22/05ME Introduction u Heat exchangers enable efficient heat transfer between two fluids at different temperatures, separated by a solid wall. u Boilers, condensers, regenerators, recuperators, preheaters, intercoolers, economizers, feedwater heaters, “radiators”, are all HXers. u Types of HXers: – Concentric tube – simple, inexpensive, easy to analyze – Shell and tube – high efficiency, expensive, common for large-scale liquid-liquid heat exchange; difficult to analyze; performance based on empirical data – Cross Flow – high-efficency, moderately expensive, common for gas-liquid heat exchange; difficult to analyze; performance based on empirical data.

1/22/05ME HXer Energy Balance Equations u Hot-side heat transfer rate u Cold-side heat transfer rate u Heat capacities u For an evaporating or condensing fluid:

1/22/05ME Log-Mean Temperature Difference (LMTD) Method of Analysis u For concentric tube hxers, u If hxer is shell-and-tube or cross flow type,

1/22/05ME Effectiveness(  )-NTU Method of Heat Exchanger Analysis u LMTD method requires an iterative procedure if only the inlet temperatures are known; in such cases, the  -NTU method is preferred. u Effectiveness (  ) is defined as u q max corresponds to a CF hxer of infinite length where the fluid with the least heat capacity experiences the maximum possible temperature change, T hi -T ci. u Thus, the actual heat transfer rate is found by

1/22/05ME Effectiveness-NTU Method, cont. u For any hxer, it can be shown that u C min /C max is equal to C h /C c or C c /C h, depending on the relative magnitudes of the fluid mass flow rates and specific heats; the number of transfer units ( NTU ) is a dimensionless parameter given by u NTU typically has values between and indicates the relative size, or heat exchange area, of the hxer.

1/22/05ME Effectiveness-NTU Method, cont. u Relations for  = f(NTU) and NTU = f(  ) are given in Tables 11.3 and 11.4, respectively, for the three types of hxers. u Figures give this same information in graphical form. u In summary, either the LMTD or  -NTU methods may be used to solve hxer problems and both will yield identical values. However, the LMTD method is best-suited for design calculations, i.e., where one outlet temperature is known and the required heat exchange area ( A ) is sought.