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UNIT II CONVECTIVE HEAT TRANSFER Course Material UNIT II CONVECTIVE HEAT TRANSFER Course material Adapted from: 1. Warren. L, McCabe, Julian,C. Smith and Peter Harriott, “Unit Operations of Chemical Engineering”, 7th Edn., McGraw Hill International Edition, NewYork 2005. 2. Holman. J.P., “Heat Transfer” , 9th Edn., Tata McGraw Hill Book Co., New Delhi, 2008. 3. R.C.Sachdeva, “Fundamentals of Engineering Heat and Mass Transfer”, 4th Edition,New Age International Publishers,2010 4. http://nptel.ac.in/courses/103103032/ - Dr. Anil Verma Dept. of Chemical Engineering,IIT Guwahati 5. www.che.utexas.edu/course/che360/lecture_notes/chapter_2.ppt CONTENTS Heat flux, average temperature of fluid stream, overall heat transfer coefficient, LMTD, individual heat transfer coefficients, relationship between individual and overall heat transfer coefficients. Concept of heat transfer by convection, natural and forced convection, application of dimensional analysis for convection, heat transfer to fluids without phase change: heat transfer coefficient calculation for natural and forced convection, heat transfer to fluids with phase change: heat transfer from condensing vapours, dropwise and film-type condensation, heat transfer coefficients calculation for film-type condensation. 2.1. Heat flux, average temperature of fluid stream Heat flux: Heat transfer calculations are based on the area of the heating surface area. The rate of heat transfer per unit area is called the heat flux. Bulk mean temperature: When a fluid is heated or cooled the temperature will vary throughout the cross section of the stream. Because of these temperature gradients through out the stream it is necessary to state what is meant by the temperature of the stream. It is the temperature that will be attained if the entire 1 fluid stream flowing across the section are withdrawn and mixed adiabatically to a uniform temperature. This is also called as average or mixing cup stream temperature. Film temperature: It is the average between the temperature of the surface and the fluid. 2.2. Overall Heat Transfer Coefficient Let us consider a plane wall of thickness xw and thermal conductivity kw. The warm fluid at a mean temperature of Th is flowing through the inside surface of the wall. The cold fluid at a mean temperature of Tc is flowing through the outside surface of the wall. The inside surface temperature is Twh and outside surface temperature is Twc. The overall heat transfer coefficient is constructed from the individual coefficients and the resistances of the wall in the following manner. (i) Overall Heat transfer Coefficient based on outside surface area The rate of heat transfer from the warm fluid to the inner surface of the wall in differential form: dq hi Th Twh (1) dAi By rearranging eqn(1), we get T T dq h wh 1 hi dAi The rate of heat transfer through the wall in differenti=al form dq k T T w wh wc (2) d AL xw By rearranging eqn(2), we get 2 T T dq wh wc xw k w d AL The rate of heat transfer from the outer surface of the wall to the cold fluid in differential form: dq ho Twc Tc (3) dAo By rearranging eqn(3), we get T T dq wc c 1 ho dAo If the eqns(1) to (3) are solved for the temperature differences and the temperature differences added, the result is 1 x 1 T T + T T + T T = T T ΔT = dQ w (4) h wh wc c wc c h c hi dAi k wd AL ho dAo Assume that the heat transfer rate is arbitrarily based on the outside area. If the eqn(4) is solved for dQ, and if both sides of the resulting equations are divided by dAo, the result is dQ T T h c (5) dAo 1 dAo xw dAo 1 dAo hi dAi k w d AL ho dAo dA r dA r o = o ; o = o dAi ri d AL r L Eqn (5) becomes dQ T T h c (6) dAo 1 ro ro ro 1 ln hi ri k w ri ho 3 From eqn (6), the overall heat transfer coefficient based on outside surface area is, 1 U o (7) 1 ro ro ro 1 ln hi ri kw ri ho (ii) Overall heat transfer coefficient based on inside surface area The rate of heat transfer from the warm fluid to the inner surface of the wall in differential form: dq hi Th Twh (1) dAi By rearranging eqn(1), we get T T dq h wh 1 hi dAi The rate of heat transfer through the wall in differential form dq k T T w wh wc (2) d AL xw By rearranging eqn(2), we get T T dq wh wc xw k w d AL The rate of heat transfer from the outer surface of the wall to the cold fluid in differential form: dq ho Twc Tc (3) dAo 4 By rearranging eqn(3), we get T T dq wc c 1 ho dAo If the eqns(1) to (3) are solved for the temperature differences and the temperature differences added, the result is 1 x 1 T T + T T + T T = T T ΔT = dQ w (4) h wh wc c wc c h c hi dAi k wd AL ho dAo Assume that the heat transfer rate is arbitrarily based on the inside area. If the eqn(4) is solved for dQ, and if both sides of the resulting equations are divided by dAi, the result is dQ T T h c (5) dAi 1 dAi xw dAi 1 dAi hi dAi k w d AL ho dAo dA r dA r i = i ; i = i dAo ro d AL r L Eqn (5) becomes dQ T T h c (6) dAi 1 ri ri ro 1 ri ln hi ri k w ri ho ro From eqn (6), the overall heat transfer coefficient based on inside surface area is, 1 U i (7) 1 ri ro 1 ri ln hi kw ri ho ro 5 Fouling factors In actual service, heat transfer surfaces do not remain clean. Scale, dirt and other solid deposits form on one or both the sides of the tubes, provide additional resistances to heat flow and reduce the overall coefficient. The effect of such deposits is taken into account as fouling factors in design calculation of heat exchangers. hdi, hdo are the fouling factors for the scale deposits on the inside and outside tube surfaces. Overall heat transfer coefficient based on outside surface area 1 U o 1 ro 1 ro ro ro 1 1 ln hi ri hd i ri kw ri hd o ho Overall heat transfer coefficient based on inside surface area 1 U i 1 1 ri ro 1 ri 1 ri ln hi hd i kw ri hd o ro ho ro 2.3. Concept of heat transfer by convection, natural and forced convection When a current or macroscopic particle of fluid crosses a specific surface, it carries with it a definite quantity of enthalpy. Such a flow of enthalpy is called convection. Convection can refer to the flow of heat associated with the movement of fluid, such as when hot air from a furnace enters a room, or to the transfer of heat from a hot surface to a flowing fluid. 6 Fig.1. Convection The two types of convection are Natural convection and forced convection. Natural convection: If the convection currents are the result of buoyancy forces generated by the differences in density and the differences in density are in turn caused by temperature gradients in the fluid mass, the action is called natural convection. Forced convection: If the convection currents are set in motion by the action of a mechanical device such as pump or agitator, the flow is independent of density gradients, it is called forced convection. 7 Fig.2.(a) Natural Convection (b) Forced Convection 2.4. Newton’s law of cooling According to Newton’s law of cooling, convective heat flux is proportional to the difference between the surface temperature and the temperature of the fluid. Q T A Q = h A (Ts –Tf) Where Q = rate of heat transfer h= heat transfer coefficient A = heat transfer area T = (Ts –Tf) = temperature difference Ts = surface temperature Tf = bulk temperature of fluid 8 2.5. Application of dimensional analysis for convection Many engineering problems can’t be solved completely by theoretical or mathematical methods. Problems of this type are common in fluid flow, heat flow. One method of attacking a problem for which no mathematical equation can be derived is that of empirical experimentation. The empirical method of obtaining an equation, is laborious, and it is difficult to organize or correlate the results so obtained into a useful relationship for calculation. There exists a method intermediate between mathematical equation and empirical equation. It is based on the fact that if a theoretical equation does exist among the variables involved in the process, that equation must be dimensionally homogeneous. It is possible to group many factors into a smaller number of dimensionless groups of variables. The groups themselves appear in the final equation. This method is called dimensional analysis. (i) Natural convection Let us consider the case of natural convection from a vertical plane wall to an adjacent fluid. Variables Symbol Dimension Length L L Fluid density ML-3 Fluid viscosity ML-1t-1 Fluid heat capacity Cp L2t-2-1 Fluid coefficient of -1 thermal expansion Acceleration due to g Lt-2 gravity Temperature difference ΔT Heat transfer coefficient h Mt-3-1 Fluid thermal conductivity k MLt-3-1 9 Number of groups = number of variables(N) – number of fundamental dimensions(m) Number of groups = 9-4 = 5 a1 b1 c1 d1 1 = L k g a2 b2 c2 d2 2 = L k g Cp a3 b3 c3 d3 3 = L k g a4 b4 c4 d4 4 = L k g ΔT a5 b5 c5 d5 5 = L k g h a1 -1 -1 b1 -3 -1 c1 -2 d1 -3 1 = (L) (ML t ) (MLt ) (Lt ) ML By equating the net dimensions of mass, length, time and temperature to zero, we get b1+c1+1 = 0 (1) a1 – b1 +c1 +d1 -3 =0 (2) -b1 -3c1 -2d1 =0 (3) -c1 = 0 (4) By solving these equations, we get c1 = 0 b1 = -1 d1 = ½ a1 = 3/2 3/2 1/2 1 = L g / Squaring of both sides L3 g 2 1 2 a2 -1 -1 b2 -3 -1 c2 -2 d2 2 -2 -1 2 = (L) (ML t ) (MLt ) (Lt ) L t By equating the net dimensions of mass, length, time and temperature to zero and solving those equations, we get 10 C p N 2 k Pr a3 -1 -1 b3 -3 -1 c3 -2 d3 -1 3 = (L) (ML t ) (MLt ) (Lt ) By equating the net dimensions of mass, length, time and temperature to zero and solving those equations, we get Lg 3 k a4 -1 -1 b4 -3 -1 c4 -2 d4 4 = (L) (ML t ) (MLt ) (Lt ) By equating the net dimensions of mass, length, time and temperature to zero and solving those equations, we get kT 4 Lg a5 -1 -1 b5 -3 -1 c5 -2 d5 -3 -1 5 = (L) (ML t ) (MLt ) (Lt ) Mt By equating the net dimensions of mass, length, time and temperature to zero and solving those equations, we get hL N 5 K Nu Combining 1, 3, 4 ; L3 2 gT N 2 Gr NNu = f(NGr, Npr) Where NNu = Nusselt number NGr = Grashof number Npr = Prandtl number 11 (ii) Forced Convection Let us consider the flow of fluid through a hot tube.
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