9. Forced Convection Correlations

Transcription

1 Part B: Heat Transfer incials in Electronics Cooling 9. Forced Convection Correlations Our rimary objective is to determine heat transfer coefficients (local and average) for different flow geometries and this heat transfer coefficient (h) may be obtained by eerimental or theoretical methods. Theoretical methods involve solution of the boundary layer equations to get the sselt number such as elained before in the revious lecture. On the other hand the eerimental methods involve erforming heat transfer measurement under controlled laboratory conditions and correlating data in dimensionless arameters. Many correlations for finding the convective heat transfer coefficient are based on eerimental data which needs uncertainty analysis although the eeriments are erformed under carefully controlled conditions. The causes of the uncertainty are many. Actual situations rarely conform comletely to the eerimental situations for which the correlations are alicable. Hence one should not eect the actual value of the heat transfer coefficient to be within better than 10%of the redicted value. 9.1 Flow over Flat Plate With a fluid flowing arallel to a flat late we have several cases arise: Flows with or without ressure gradient Laminar or turbulent boundary layer Negligible or significant viscous dissiation (effect of frictional heating) Flow with Zero essure Gradient and Negligible Viscous issiation When the free-stream ressure is uniform the free-stream velocity is also uniform. Whether the boundary layer is laminar or turbulent deends on the Reynolds number Re X (ρu /µ) and the shae of the solid at entrance. With a shar edge at the leading edge the boundary layer is initially laminar but at some distance downstream there is a transition region and downstream of the transition region the boundary layer becomes turbulent. For engineering alications the transition region is usually neglected and it is assumed that the boundary layer becomes turbulent if the Reynolds number Re X is greater than the critical Reynolds number Re cr. A tyical value of for the critical Reynolds number is generally acceted. The viscous dissiation and high-seed effects can be neglected if 1/2 Ec<1. The Eckert number Ec is defined as Ec u 2 /C (T S -T ) with a rectangular late of length L in the direction of the fluid flow Laminar Boundary Layer (Re ) With heating or cooling starting from the leading edge the following correlations are recommended. Note: in all equations evaluate fluid roerties at the film temerature (T f ) defined as the arithmetic mean of the surface and free-stream temeratures unless otherwise stated. T f (T S + T )/2 75

2 Part B: Heat Transfer incials in Electronics Cooling Flow with uniform surface temerature (T S constant) - Local heat transfer coefficient The sselt number based on the local convective heat transfer coefficient is eressed as 1/ 2 X f Re X (9.1) The eression of ƒ deend on the fluid andtl number For liquid metals with very low andtl number liquid metals ( 0.05) 1 / 2 f For 0.6 < < 50 For very large andtl number f f For all andtl numbers; Correlations valid develoed by Churchill (1976) and Rose (1979) are 1 / Re (9.2) 2 / 3 1 / Average heat transfer coefficient The average heat transfer coefficient can be evaluated by erforming integration along the flat late length if andtl number is assumed constant along the late the integration yields to the following result: 2 (9.3) Flow with uniform heat flu (q // constant) - Local heat transfer coefficient Churchill and Ozoe (1973) recommend the following single correlation for all andtl numbers X Re 1 + 1/ 2 X / 3 1/ 2 1/ 4 (9.4) 76

3 Part B: Heat Transfer incials in Electronics Cooling Turbulent Boundary Layer (Re > ) With heating or cooling starting from the leading edge the following correlations are recommended. - Local heat transfer coefficient Re cr < Re Re 4 / 5 1/ 3 0 Re > / Re (ln Re ) (9.5) (9.6) Equation 9.6 is obtained by alying Colburn s j factor in conjunction with the friction factor suggested by Schlichting (1979). With turbulent boundary layers the correlations for the local convective heat transfer coefficient can be used for both uniform surface temerature and uniform heat flu. - Average heat transfer coefficient If the boundary layer is initially laminar followed by a turbulent boundary layer some times called mied boundary layer the Following correlations for 0.6 < < 60 are suggested: Re cr < Re 10 7 ( Re 4 / 5 871) 1/ 3 Re >10 7 [ Re (ln Re ) ] (9.7) (9.8) Unheated Starting Length Uniform Surface Temerature & > 0.6 If the late is not heated or (cooled) from the leading edge where the boundary layer develos from the leading edge until being heated at o as shown in Figure 9.1 the correlations have to be modified.. Leading edge δ δ T o Figure 9.1 Heating starts at o - Local heat transfer coefficient 77

4 Part B: Heat Transfer incials in Electronics Cooling Laminar flow (Re < Re cr ) Re o 1 1 / 2 3 / 4 (9.9) Turbulent flow (Re > Re cr ) Re o 1 4 / 5 9 / 10 3 / 5 1 / 9 (9.10) - Average heat transfer coefficient over the Length (L o) Laminar flow (Re L < Re cr ) h L o k L o h 2 1 ( Re L o 1 / 2 L o 1 / L) L 3 / 4 o 1 L 3 / 4 2 / 3 (9.11) Evaluate h L from Equation 9.9. Turbulent flow (Re L > Re cr ) h L 4 / 5 3 / 5 o Re L 1 L L o 9 /10 o 1 L 1.25 h L o 1 L o 9 /10 8 / 9 k (9.12) Evaluate h L from Equation Eamle 9.1: Eerimental results obtained for heat transfer over a flat late with zero ressure gradients yields 0.9 1/ Re Where this correlation is based on (the distance measured from the leading edge). Calculate the ratio of the average heat transfer coefficient to the local heat transfer coefficient. Schematic: 78

5 Part B: Heat Transfer incials in Electronics Cooling Solution: It is known that Local heat transfer coefficient h k h k Re The average heat transfer coefficient is: h k 0.04k k hd 0.04k Re 0.9 h 0.9 1/ u v u v u v 0.1 d d The ratio of the average heat transfer coefficient to the local heat transfer coefficient is: h 1 h 0.9 Eamle 9.2: Air at a temerature of 30 o C and 6 kpa ressure flow with a velocity of 10 m/s over a flat late 0.5 m long. Estimate the heat transferred er unit width of the late needed to maintain its surface at a temerature of 45 o C. Schematic: P 6 ka u 10 m/s T 30 o C T S 45 o C L0.5 m 79

6 Part B: Heat Transfer incials in Electronics Cooling Solution: oerties of the air evaluated at the film temerature T f 75/ o C From air roerties table at 37.5 o C: ν m 2 /s. k W/m. o C C J/kg. o C Note: These roerties are evaluated at atmosheric ressure thus we must erform correction for the kinematic viscosity ν) act. ν) atm. (1.0135/0.06) m 2 /s. Viscous effect check Ec u 2 /C (T S -T ) (10) 2 /( ) It roduce 1/2 Ec<1 so that we neglect the viscous effect Reynolds number check Re L u L / ν / ( ) The flow is Laminar because Re L Using the Equation / 2 L Re L ( ) 1/2 (0.7055) 1/ Now the average heat transfer coefficient is hl L k h / W/m 2. o C Then the total heat transfer er unit width equals q h L (T S -T ) (45 30) W/m Eamle 9.3: Water at 25 o C is in arallel flow over an isothermal 1-m long flat late with velocity of 2 m/s. Calculate the value of average heat transfer coefficient. Schematic: u 2 m/s T 25 o C Solution: Assumtions L1 m 80

7 Part B: Heat Transfer incials in Electronics Cooling - neglect the viscous effect - The roerties of water are evaluated at free stream temerature From water roerties table at 25 o C: ν m/s. k 0.613W/m. o C 5.83 C 4180 J/kg. o C Reynolds number check Re L u L / ν 2 1 / ( ) The flow is mied because Re L By using the Equation 9.7 (0.037 Re / 5 871) 6 [0.037( ) 871](5.83) h L L k The average heat transfer coefficient is 2 h L 4110 W/m K 9.3 Flow over Cylinders Sheres and other Geometries The flow over cylinders and sheres is of equal imortance to the flow over flat late they are more comle due to boundary layer effect along the surface where the free stream velocity u brought to the rest at the forward stagnation oint (u 0 and maimum ressure) the ressure decrease with increasing is a favorable ressure gradient (d/d<0) bring the ressure to minimum then the ressure begin to increase with increasing by adverse ressure gradient (d/d>0) on the rear of the cylinder. In general the flow over cylinders and sheres may have a laminar boundary layer followed by a turbulent boundary layer. The laminar flow is very weak to the adverse ressure gradient on the rear of cylinder so that the searation occurs at θ 80 o and caused wide wakes as show in Figure9.2.a. The turbulent flow is more resistant so that the searation occurs at θ 120 o and caused narrow wakes as show in Figure9.2.b. u T P searation θ wide wakes 4 / 5 u T P θ searation narrow wakes stagnation oint (a) stagnation oint (b) Figure 9.2 Flow over cylinder Because of the comleity of the flow atterns only correlations for the average heat transfer coefficients have been develoed. 81

8 Part B: Heat Transfer incials in Electronics Cooling Cylinders The emirical relation reresented by Hilert given below in Equation 9.13 is widely used where the constants c m is given in Table 9.1 all roerties are evaluated at film temerature T f. 1/ 3 c Re m Table 9.1 Constants of Equation 9.13 at different Reynolds numbers Re c m Other correlation has been suggested for circular cylinder. This correlation is reresented by Zhukauskas given below in Equation 9.14 where the constants c m are given in Table 9.2 all roerties are evaluated at Free stream temerature T ecet S which is evaluated at T S which is used in limited andtl number 0.7 < < 500 If 10 n 0.37 If > 10 n 0.36 c Re m n Table 9.2 Constants of Equation 9.14 at different Reynolds numbers Re c m For entire ranges of Re as well as the wide ranges of andtl numbers the following correlations roosed by Churchill and Bernstein (1977): Re > 0.2. Evaluate roerties at film temerature T f : S 1/ 4 (9.13) (9.14) Re / Re / / 4 Re / 8 4 / 5 (9.15) Re

9 Part B: Heat Transfer incials in Electronics Cooling 1 / Re / 3 1 / 4 Re Re / 2 1/ Re / 3 1/ For flow of liquid metals use the following correlation suggested by Ishiguro et al. (1979): 1 Re d d.125(re ) 1 1 / 2 (9.16) (9.17) (9.18) Sheres Boundary layer effects with flow over circular cylinders are much like the flow over sheres. The following two correlations are elicitly used for flows over sheres 1. Whitaker (1972): All roerties at T ecet s at T s 3.5 Re d s ( 0.4 Re 1/ Re 2 / ) 2. Achenbach (1978): All roerties at film temerature 100 Re d / 5 µ µ S 5 ( 0.25 Re Re 8 / ) 1/ / 4 (9.20) (9.21) Re d Re Re Re (9.22) For Liquid Metals convective flow eerimental results for liquid sodium Witte (1968) roosed Re d / (Re ) (9.23) Other Geometries For geometries other than cylinders and sheres use Equation 9.24 with the characteristic dimensions and values of the constants given in the Table 9.3 for different geometries all roerties are evaluated at film temerature T f. 1/ 3 c Re m (9.24) Equation 9.24 is based on eerimental data done for gases. Its use can be etended to fluids with moderate andtl numbers by multilying Equation 9.24 by

10 Part B: Heat Transfer incials in Electronics Cooling Table 9.3 Constants for Equation 9.24 for non circular cylinders eternal flow Geometry Re C m u u u u u Eamle 9.4: Atmosheric air at 25 o C flowing at velocity of 15 m/s. over the following surfaces each at 75 o C. Calculate the rate of heat transfer er unit length for each arrangement. a) A circular cylinder of 20 mm diameter b) A square cylinder of 20 mm length on a side c) A vertical late of 20 mm height Schematic: u u u d (a) (b) (c) Solution: From the film temerature Air roerties are: ν m/s. k 0.028W/m. o C T f (25+75)/2 50 o C Calculation of Reynolds number for all cases have the same characteristic length20mm Re u L / ν / By using Equation 9.13 for all cases 1/ 3 c Re m 84

11 Part B: Heat Transfer incials in Electronics Cooling Case (a) a circular cylinder From Table 9.1 C and m ( ) ( ) 1/ 3 The average heat transfer coefficient is k o h 97.7 W/m. C 0.02 The total heat transfer er unit length is q / 97.7 (π 0.02) W/m Case (b) a square cylinder From Table 9.3 C and m ( ) ( ) The average heat transfer coefficient is k o h W/m. C 0.02 The total heat transfer er unit length is q / (4 0.02) W/m Case (c) a vertical late From Table 9.3 C and m ( ) ( ) The average heat transfer coefficient is k o h W/m. C 0.02 The total heat transfer er unit length is q / (2 0.02) W/m / 3 1/ Heat Transfer across Tube Banks (Heat Echangers) Heat transfer through a bank (or bundle) of tubes has several alications in industry as heat echanger which can be used in many alications. When tube banks are used in heat echangers two arrangements of the tubes are considered aligned and staggered as shown in Figure 9.3. If the sacing between the tubes is very much greater than the diameter of the tubes correlations for single tubes can be used. Correlations for flow over tube banks when the sacing between tubes in a row and a column are not much greater than the diameter of the tubes have been develoed for use in heat-echanger alications will be discussed as follows. 85

12 Part B: Heat Transfer incials in Electronics Cooling N L S L S S L T u N T S T T u S T row 1 row 2 row 3 row 4 (a) row 1 row 2 row 3 row 4 Figure 9.3 Arrangements of the tubes (a) In-line arrangement (b) staggered arrangement For the average convective heat transfer coefficient with tubes at uniform surface temerature eerimental results carried by Zukauskas (1987) recommended the following correlation: P m n 0.25 c( a / b) Re (/ s ) (9.25) a S T /; b S L /. S T Transverse itch S L Lateral itch tube diameter All roerties are evaluated at the arithmetic mean of the inlet and eit temeratures of the fluid (T i + T o )/2 ecet S which is evaluated at the surface temerature T S. The values of the constants c m and n are given in Table 9.4 for in-line arrangement and in Table 9.5 for staggered arrangement. The maimum average velocity between tubes is used to calculate Re. The maimum velocity can be calculated as follows: ST For in-line arrangement: u ma u ST ST + S > 2 For Staggered arrangement: If ST u ma u S T (b) Where If not u S ma S 0.5S ( S ) 2 L T u 2 S 2 T + 1/ 2 Table 9.4 In-Line arrangement values of constants in Equation 9.25 ( 0 in all cases) Re C m n

13 Part B: Heat Transfer incials in Electronics Cooling Table 9.5 Staggered arrangement values of constants in Equation 9.25 Re c P m n The temerature of the fluid varies in the direction of flow and therefore the value of the convective heat transfer coefficient (which deends on the temerature-deendent roerties of the fluid) also varies in the direction of flow. It is a common ractice to comute the total heat transfer rate with the assumtion of uniform convective heat transfer coefficient. With such an assumtion of uniform convective heat transfer coefficient uniform surface temerature and constant secific heat the heat transfer rate to the fluid is related by the heat balance. T o + T i q m C P ( T o T i ) ha TS (9.26) 2 T i inlet free stream temerature T o outlet free stream temerature m out side tubes gaseous flow rate ρ i (N T S T L)u A Total heat transfer area N T N L πl N T number of tubes in transverse direction N L number of tubes in lateral direction L length of tube er ass. essure dro across tube banks: essure dro is a significant factor as it determines the ower required to move the fluid across bank. The ressure dro for flow gases over a bank may be calculated with Equation For in-line: 2 f / G ρ 2 ma N T µ S µ ressure dro in ascals. ƒ / friction factor is given by Jacob G ma mass velocity at minimum flow rate kg/m 2.s ρ density evaluated at free stream conditions kg/m 3 N T number of transverse rows. µ S fluid viscosity evaluated at surface temerature. µ average free stream viscosity (9.27) 87

14 Part B: Heat Transfer incials in Electronics Cooling f / S T 0.08S / L / S L Re 0.15 ma (9.28) For staggered: f / ST 1.08 Re 0.16 ma (9.29) Eamle 9.5: A heat echanger with aligned tubes is used to heat 40 kg/sec of atmosheric air from 10 to 50 C with the tube surfaces maintained at 100 C. etails of the heat echanger are iameter of tubes 25 mm mber of columns (N T ) 20 Length of each tube 3 m S L S T 75 mm -etermine the number of rows (N L ) required Solution: oerties of atmosheric air at average air temerature (T i + T o )/2 30 C. ρ kg/ m 3 C P 1007 J/ kg. o C k W/ m. o C µ N.s/ m At surface temerature T s 100 o C s At inlet free stream temerature T i 10 o C ρ i 1.24 kg/ m 3 To find u m ρ i (N T S T L)u ( ) u u m/s For in-line arrangement: u ST u ST ( ) ma 88

15 Part B: Heat Transfer incials in Electronics Cooling m/ s Reynolds number based on maimum velocity Re ρu ma /µ /( ) From Table C 0.27 m 0.63 n 0.36 From Equation ( ) 0.63 (0.7066) 0.36 (0.7066/0.6954) h / k The average heat transfer coefficient is h / W/m 2.k From Equation 9.26.The Total heat transfer is T o + T q m C P ( T o T i ) ha TS (50-10) A [100-30] i Total heat transfer area mber of rows (N L ) A m 2 A N T N L πl N L (π ) N L 43 rows 9.5 Heat Transfer with Jet Imingement The heat transfer with jet imingement on a heated (or cooled) surface results in high heat transfer rates and is used in a wide range of alications such as cooling of electronic equiments. Usually the jets are circular with round nozzle of diameter d or rectangular with width w. They may be used individually or in an array. The jets may iminge normally to the heated surface as shown in Figure 9.4 or at an angle. The liquid coolant under ressure inside a chamber is allowed to ass through a jet and directly into the heated surface there are two modes of oeration ossible with liquid jet imingement namely single hase cooling and two hases cooling. And the jet may be free or submerged. If there is no arallel solid surface close to the heated surface the jet is said to be free as shown in Figure 9.4 while Jets may be submerged if the cavity is comletely filled with the fluid (from the nozzle into a heated surface). In this lecture only single free jets (round or rectangular) iminging normally to the heated surface are considered. 89

16 Part B: Heat Transfer incials in Electronics Cooling d H Figure 9.4 Jet iminge normally to the heated surface Cooling analysis: Free single hase jet imingement cooling is affected with many variables such as: - Jet diameter (d) - Fluid velocity (v) - Jet to heated surface distance (H ) - Size of heated surface area (L L) - Coolant roerties The average heat transfer coefficient correlation is given by Jigi and agn L L 3.84 Re d d The roerties are evaluated at mean film temerature (T S + T )/2 This correlation is eerimented for FC-77 and water and also valid for 3< H/d <15-3< H/d <15 - d to mm - v < 15 m/s - small surface dimensions L<12.7 mm (microelectronic devices) (9.30) Eamle 9.6: A single hase free jet imingement nozzle is laced in the center of an electronic heated surface 1212 mm 2 and the heated surface is laced at 4 mm from the jet. The working medium is FC-77 assing through 1mm tube diameter at a rate of kg/s. To cool the late if the suly coolant is at 25 o C and the heat load is 20 W etermine the average heat transfer coefficient and the surface temerature of the heated surface. Solution: To get the roerties of the FC-77 we need to assume the surface temerature: let the surface temerature equal 45 o C as a first aroimation. T f ( )/2 35 o C From FC-77 roerty tables at 35 o C: ρ 1746 kg/m 3 µ N.s/m 2 k W/m. o C 20.3 We must check on H/d ratio: H/d 4/1 Calculation of Reynolds number 90

17 From Equation 9.30 Part B: Heat Transfer incials in Electronics Cooling Re ρvd / µ 4m ' / πdµ / π L L 3.84 Re d d 3.84 (15942) 0.5 (20.3) 0.33 ( /1 +1) hl Knowing that l k And so the average heat transfer coefficient is: h ( )/ ( ) W/m 2.k But since q h A T Then the temerature difference is: T 20/ ( ) o C Therefore the surface temerature is: T S o C For more accuracy we may make another trial at new film temerature and reach more accurate results Case Study The use of iminging air jets is roosed as a means of effectively cooling high-ower logic chis in a comuter. However before the technique can be imlemented the convection coefficient associated with jet imingement on a chi surface must be known. esign an eeriment that could be used to determine convection coefficients associated with air jet imingement on a chi measuring aroimately 10 mm by 10 mm on a side. 9.6 Internal Flows (Inside Tubes or ucts) The heat transfer to (or from) a fluid flowing inside a tube or duct used in a modern instrument and equiment such as laser coolant lines comact heat echanger and electronics cooling(heat ie method). Only heat transfer to or from a single-hase fluid is considered. The fluid flow may be laminar or turbulent the flow is laminar if the Reynolds number (u m H /) is less than 2300 (Re 2300) based on the tube hydraulic diameter ( H 4A / P) where A P is the cross sectional area and wetted erimeter resectively and u m is average velocity over the tube cross section. Also the hydraulic diameter should be used in calculating sselt number. And If the Reynolds number is greater than 2300 the flow is turbulent (Re >2300) Fully eveloed Velocity ofiles When a fluid enters a tube from a large reservoir the velocity rofile at the entrance is almost uniform because the flow at the entrance is almost inviscid while the effect of the viscosity increase with the length of tube (in -direction) which have an effect on the shae of the velocity rofile as shown in Figure

18 Part B: Heat Transfer incials in Electronics Cooling At some location downstream of the ie inlet the boundary layer thickness ( reaches its maimum ossible value which is equal to the radius of the tube at this oint and the velocity rofile does not change. The distance from the entrance to this oint is called fully develoed region or entrance region and it is eressed as the fully develoing length (L ) this length deends on the flow which may be laminar or turbulent. Through the develoing length (L ) the heat transfer coefficient is not constant and after the develoing length the heat transfer coefficient is nearly constant as shown in Figure 9.5. h ( W/m 2.K ) u m X R δ L (Fully develoing length) Figure 9.5 Flow inside the ie Fully develoed flow The fully develoing length is given by: L / 0.05Re -For laminar flow And L / 10 -For turbulent flow Heat Transfer Correlations Through the entrance region and after this region (fully develoed flow) both regions have different heat transfer correlations given in Table

19 Part B: Heat Transfer incials in Electronics Cooling Table 9.6 Summery for forced convection heat transfer correlations - internal flow Correlations Conditions Sieder and Tate (1936) - Laminar entrance region - uniform surface temerature Re (9.31) µ - L/ < Re µ 1.86 / 8 µ S L µ S < < < µ /µ S < (9.32) - Laminar fully develoed - Uniform surface temerature (9.33) - Laminar fully develoed - Uniform Heat Flu ittus Boelter (1930) 4 / 5 n 0.023Re d (9.34) Sleicher and Rouse (1976) Re m f S + m Re f S (9.35) (9.36) - Turbulent flow fully develoed L/ 10 - n 0.4 for heating (T s > T m ) and - n 0.3 for cooling (T s < T m ). - For liquid metals «1 - Uniform surface temerature - For liquid metals «1 - Uniform heat flu oerties of the fluid for each equation: For Equation 9.31 at the arithmetic mean of the inlet and eit temeratures (T mi +T mo )/2 For Equation 9.32 Equation 9.33 Equation 9.34 and at the mean temerature T m For Equation 9.35 and Equation 9.36 the subscrits m f and s indicate that the variables are to be evaluated at the mean temerature T m film temerature T f (arithmetic mean of the mean temerature and surface temeratures (T S +T m )/2 and surface temerature resectively Variation of Fluid Temerature(T m ) in a Tube By taking a differential control volume as shown in Figure 9.6 considering that the fluid enters the tube at T mi and eits at T mo with constant flow rate m ' convection heat transfer occurring at the inner surface (h) heat addition by convection q neglecting the conduction in the aial direction and assuming no work done by the fluid. Then alying heat balance on the control volume we can obtain an equation relating the surface temerature at any oint. 93

20 Part B: Heat Transfer incials in Electronics Cooling T m d dt m T mi T mo X dq Figure9.6 differential control volume inside tube Case 1: - At uniform surface temerature T S constant dq m ' C dt m (9.37) h (Pwd) (T S T m ) (9.38) Where P w out side tube erimeter L By equating the both Equations dt T T T T i X i m d S dt d hp m C T m d T d d T T ( T T w S m m T d T d hpw ( T ) m C hpw m C By integration of Equation 9.40 as shown bellow: TX d T hpw d T m C 0 d T Pw T m C 1 d 0 ) h d (9.39) (9.40) 94

21 Part B: Heat Transfer incials in Electronics Cooling 95 m C h P i w i w e T T m C h P T T ln

22 Part B: Heat Transfer incials in Electronics Cooling At L w h L T o m C T i e So that the temerature distribution along the tube is as shown in Figure 9.7 (9.41) temerature T S constant T T S -T m T o T i T mo T mi X L Figure 9.7 Temerature distributions along the tube at uniform surface temerature The total heat transfer is: But also T mo - T mi T i T o q t m C ( T mo T mi ) (9.42) So that the Equation 9.42 can be written as follows q m C ( T t i T o ) (9.43) Also for total heat transfer by average heat transfer coefficient q hlp ( T T ) From Equations roduce: m C T t average i w o S m w average ( T T ) hlp ( T ) m C ( Ti To ) hlp w average (9.44) (9.45) Where T average (T S -T m ) average Substitute the Equation 9.41 in Equation 9.45 roduces: Ti To Taverage Ti ln To T average called logarithmic mean temerature difference (LMT) (9.46) 96

23 Part B: Heat Transfer incials in Electronics Cooling Case 2: - At uniform heat flu q // constant From Equation 9.37 And dq m ' C dt m dq q // P w d (9.47) Combining both Equations roduces: // dtm q Pw ' d m C By integration: Tm // q Pw dtm d ' T m C mi 0 // qpw T m - T mi ' mc And q // h (T S T m ) T S T m q // / h constant along the tube length (9.48) (9.49) (9.50) From the Equations the temerature distribution as shown in Figure 9.8. Temerature T S T mo T m T mi Entrance Region L Fully develoed flow Figure 9.8 Temerature distributions along the tube at uniform heat flu For along the tube length // q P T mo T mi + ' m C w L 97