Heat transfer to or from a fluid flowing through a tube

Transcription

1 Heat tranfer to or from a fluid flowing through a tube R. Shankar Subramanian A common ituation encountered by the chemical engineer i heat tranfer to fluid flowing through a tube. Thi can occur in heat exchanger, boiler, condener, evaporator, and a hot of other proce equipment. Therefore, it i ueful to know how to etimate heat tranfer coefficient in thi ituation. We can claify the flow of a fluid in a traight circular tube into either laminar or turbulent flow. It i aumed from hereon that we aume fully developed incompreible, Newtonian, teady flow condition. Fully developed flow implie that the tube i long compared with the entrance length in which the velocity ditribution at the inlet adjut itelf to the geometry and no longer change with ditance along the tube. Reynold number The value of the Reynold number permit u to determine whether the flow i laminar or turbulent. We define the Reynold number a follow. DV ρ Reynold number Re = = μ DV ν Here, D i the inide diameter of the tube (or pipe), V i the average velocity of the fluid, ρ i the denity of the fluid and μ i it dynamic vicoity. It i common to ue the kinematic vicoity ν = μ/ ρ in defining the Reynold number. Another common form involve uing the ma flow rate m intead of the average velocity. The ma flow rate i related to the volumetric π 2 flow rate Q via m= ρq, and we can write Q= D V. Therefore, the Reynold number alo 4 can be defined a 4m Re = πμd The flow in a commercial circular tube or pipe i uually laminar when the Reynold number i below 2,300. In the range 2,300 < Re < 4, 000, the tatu of the flow i in tranition and for Re > 4, 000, flow can be regarded a turbulent. Reult for heat tranfer in the tranition regime are difficult to predict, and it i bet to avoid thi regime in deigning heat exchange equipment. By the way, turbulent flow i inherently unteady, being characterized by time-dependent fluctuation in the velocity and preure, but we uually average over thee fluctuation and define time-moothed or time-average velocity and preure; thee time-moothed entitie can be teady or time-dependent (on a time cale much larger than that of the fluctuation), and here we only focu on teady condition when we dicu either laminar or turbulent flow. 1

2 Principal difference between heat tranfer in laminar flow and that in turbulent flow In dicuing heat tranfer to or from a fluid flowing through a traight circular tube, it i ueful to ditinguih between the axial or main flow direction, and the direction that lie in a plane perpendicular to the tube axi. In that plane, tranvere heat flow can be broken into radial and azimuthal component. The principal difference between laminar and turbulent flow, a far a heat tranfer i concerned, i that an additional mechanim of heat tranfer in the radial and azimuthal direction become available in turbulent flow. Thi i commonly termed eddy tranport and i intene, providing much better tranfer of energy acro the flow at a given axial poition than in laminar flow, wherein conduction i typically the only mechanim that operate in the tranvere direction (an exception occur when there are econdary flow in the tranvere direction, uch a in coiled tube). Another difference worthwhile noting i the extent of the thermal entrance region in which the tranvere temperature ditribution become fully developed. Thi region i relatively hort in turbulent flow (preciely becaue of the intene turbulent tranvere tranport of energy), wherea it tend to be long in laminar flow. Heat tranfer correlation, baed on experimental reult, are typically divided into thoe applicable in the thermal entrance region, and thoe that apply in the fully developed region. In the cae of laminar flow, it i important to be aware of thi ditinction, and normally a laminar flow heat exchanger i deigned to be hort, to take advantage of relatively high heat tranfer rate that are achievable in the thermal entrance region. In the cae of turbulent flow, the thermal entrance region i hort, a noted earlier, and typically heat tranfer occur motly in the fully developed region. Therefore, turbulent heat tranfer correlation are commonly provided for the latter region. Laminar heat tranfer correlation A variety of correlation are in ue for predicting heat tranfer rate in laminar flow. From dimenional analyi, the correlation are uually written in the form ( Re,Pr, ) Nu = f hd μc where Nu = i the Nuelt number, f i ome function, and Pr p ν = = i the Prandtl k k α number. Here, h i the heat tranfer coefficient, k i the thermal conductivity of the fluid, and C p i the pecific heat of the fluid at contant preure. A you can ee, the Prandtl number can be written a the ratio of the kinematic vicoity ν to the thermal diffuivity of the fluid α. The ellipe in the right ide of the above reult tand for additional dimenionle group uch a L/ D, which i the ratio of the tube length to it diameter, and other group that we ll dicu a they occur. A we noted before, efficient heat tranfer in laminar flow occur in the thermal entrance region. A reaonable correlation for the Nuelt number wa provided by Sieder and Tate. 2

3 1/ /3 1/3 D μ b 1.86 Re Pr L μw Nu = You can ee that a the length of the tube increae, the Nuelt number decreae a. Thi doe not, however, imply that the Nuelt number approache zero a the length become large. Thi i becaue the Sieder-Tate correlation only applie in the thermal entrance region. In long tube, wherein mot of the heat tranfer occur in the thermally fully-developed region, the Nuelt number i nearly a contant independent of any of the above parameter. When the boundary condition at the wall i that of uniform wall temperature, Nu If intead the flux of heat at the wall i uniform, Nu 4.36, but in thi cae we already know the heat flux and a heat tranfer coefficient i not needed. Remember that the purpoe of uing a heat tranfer coefficient i to calculate the heat flux between the wall and the fluid. In the cae of uniform wall flux, we can ue an energy balance directly to infer the way in which the bulk average temperature of the fluid change with ditance along the axial direction. μ Notice that a ratio b appear in the above laminar flow heat tranfer correlation. We have μw defined μ a the vicoity of the fluid. The ubcript b and w tand for bulk and wall, repectively. We know that the bulk temperature of the fluid will change along the tube. The wall temperature may be contant, or it may vary along the length of the tube. In all cae, we can ue an arithmetic value of the average between the extreme value that occur in the ytem. Becaue the exponent (0.14) i mall, the effect of thi term on the Nuelt number i not large it i only a mall correction, and thi averaging i quite jutified. In fact, for all the other phyical propertie uch a denity, thermal conductivity, and pecific heat, we hould etimate value at the average temperature of the fluid between the inlet and outlet. The Reynold and Prandtl number are raied to the ame power in the laminar flow correlation. Therefore, we can write the correlation a 1/ / /3 D μ 1/3 D μ b b ( ) ( ) Nu = 1.86 Re Pr = 1.86 Pe L μw L μw where we have introduced a new group Pe called the Peclet number. L 1/3 Pe DV ν DV = Re Pr = = ν α α The Peclet number Pe play a role in heat tranfer that i imilar to that of the Reynold number in fluid mechanic. Recall that the phyical ignificance of the Reynold number i that it repreent the ratio of inertial force to vicou force in the flow, or equivalently, the relative importance of convective tranport of momentum compared with molecular tranport of momentum. Thu, the Peclet number tell u the relative importance of convective tranport of thermal energy when compared with molecular tranport of thermal energy (conduction). 3

4 The author of the textbook recommend the following laminar flow heat tranfer correlation from a book by D.K. Edward, V.E. Denny, and A.F. Mill for the average Nuelt number for a tube of length L. Nu average D h 0.065Re Pr averaged = = L k D RePr L 2/3 Unlike the correlation of Sieder and Tate, thi reult can be ued for hort or long tube. Note that a the length become very large, Nu 3.66, which i the reult for a uniform wall average temperature when the temperature field i fully developed. The textbook alo provide ueful information about entrance length. For example, the hydrodynamic entrance length L ef for the friction factor to decreae to within 5% of it value for fully developed laminar flow condition i given a L ef D 0.05 Re Likewie, if the velocity profile in laminar flow i fully developed and we then apply a uniform wall temperature boundary condition, the thermal entrance length can be etimated from L eh D Re Pr When both the velocity and temperature field develop with ditance imultaneouly, the problem i more involved. Turbulent flow The entrance length are much horter for turbulent flow, becaue of the additional tranport mechanim acro the cro ection. Thu, typical hydrodynamic entrance length in turbulent flow are tube diameter, and the thermal entrance length are even maller. Therefore, for L mot engineering ituation wherein 50 D, we ue correlation for fully developed condition. Correlation for turbulent flow are claified baed on whether the interior wall of the tube i mooth or whether it i rough. Smooth tube The earliet correlation for turbulent heat tranfer in a mooth tube are due to Dittu and Boelter, McAdam, and Colburn. A common form to be ued for fluid with Pr > 0.5 i 4

5 Nu = Re Pr The uual recommendation i to ue thi correlation for Re > 10,000, but in practice it i ued even when the flow i in tranition between laminar and turbulent flow for lack of better correlation. A modern correlation that i lightly more accurate i recommended in the textbook for your ue. ( f )( ) /8 Re 1,000 Pr Nu = / 8 Pr 1 1/2 2/3 ( f ) ( ) 6 Mill ugget uing thi correlation for Reynold number between 3,000 and 10. Of coure, it hould not be ued if Pr =1, but there are no fluid with that precie value of the Prandtl number. For low Prandtl number liquid metal, the textbook provide pecial correlation to be ued for uniform wall temperature and uniform wall flux boundary condition. Phyical propertie to be ued in thee correlation are evaluated at the average of the inlet and exit temperature of the fluid. The friction factor f i the Darcy friction factor, and you can ue Petukhov formula for evaluating it. f = 1 ( ) 0.790ln Re Thi reult i good for turbulent flow in mooth pipe for Re Rough tube and pipe In the cae of commercial pipe, roughne of the interior urface i inevitable, wherea drawn tube tend to be le rough. The extent of roughne depend on the nature of the urface. Mill provide a dicuion of heat tranfer in turbulent flow in rough pipe in Section The heat tranfer rate i predicted in thi cae by uing a group called the Stanton number Nu Nu St = =. Re Pr Pe f St = 8 1/2 f g( h +, Pr) where the friction factor f i calculated uing 5

6 f k / R 5.02 k / R 13 = 2.0 log10 log Re 7.4 Re 2 In the above correlation, k i known a the equivalent and grain roughne Value of k for a variety of pipe, tube, and other type of urface can be found in Table 4.8 in the ( ) textbook. The ymbol R repreent the inide radiu of the pipe. The function g h +,Pr i tabulated in Table 4.9, but you will firt need to convert h to h + which i dimenionle. The ymbol h tand for the average height of protruion from the urface. For equivalent and grain roughne, we can ue h= k. For a pipe, the relationhip between the dimenionle quantitie (+ variable) and the phyical variable i given in Equation (4.140a). For example, + Vk f k = ν 8 o that 1/2 + Vh f h = ν 8 1/ 2 Recall that V i the average velocity of flow, and Mill ue the ymbol quantity in Equation (4.140a). u b to denote thi Non-circular cro-ection To handle non-circular cro-ection uch a annular, triangular, rectangular, and the like, we ue the concept of hydraulic diameter D. Thi i defined a h D h = 4A P where A i the cro-ectional area and P i the wetted perimeter. For a circular tube of 2 diameter D, A = π D, and P= π D, o that the above definition yield Dh = D. For turbulent 4 flow in non-circular cro-ection, we can ue the correlation for circular tube. The hydraulic diameter i ued in place of the diameter in thee correlation. Reult for fully developed heat tranfer in laminar flow are given in Table 4.5 for a variety of cro-ection, and a correlation for flow between parallel plate i given in Equation (4.51) of the text. 6

7 Phyical property variation The Sieder-Tate correlation for laminar flow contain a correction for the variation of vicoity with temperature. The other correlation given here do not contain an explicit correction. Mill dicue how to accommodate property variation at the end of Section 4.2, on page 300. The approach i different for liquid and gae. For liquid, a correction i made to the value of the friction factor and that of the Nuelt number by multiplying the value calculated from the correlation uing the bulk fluid propertie (evaluated at an arithmetic average temperature between the inlet and outlet temperature for the tream) by a uitable factor. Thi factor, for the Nuelt number correction, i a vicoity ratio n μ, wherein the ubcript and b refer to the urface and bulk, repectively. In the cae of μb the bulk, an arithmetic average between the inlet and the outlet i to be ued, and if the urface temperature varie from the inlet to the outlet, an arithmetic average i to be ued a well. Likewie, the friction factor correction i made by multiplying the value obtained uing bulk m μ average temperature by. Suitable value for ue in thee correction are given in Table μb 4.6. In the cae of gae, the correction factor for the Nuelt number i T friction factor i Tb m. T Tb n and that for the 7