An alternative approach to evaluate the average Nusselt number for mixed boundary layer conditions in parallel flow over an isothermal flat plate

Transcription

1 An alternative approach to evalate the average Nsselt nber for ied bondary layer conditions in parallel flo over an isotheral flat plate Viacheslav Stetsyk, Krzysztof J. Kbiak, ande i and John C Chai Abstract In this paper, e present an alternative approach to evalate the average Nsselt nber for ied bondary layer conditions in parallel flo over an isotheral flat plate. his approach can be sed regardless of the critical Reynolds nber here the flo transitions fro lainar flo to trblent flo. his approach is siple and ses graphical visalisation of the physical sitation. his shold assist coprehension and retention. It tilises the average qantity for the lainar bondary layer and the average vale for trblent bondary layer to obtain the average qantities for ied bondary layers ithot the need to perfor the sal integration. It can easily be incorporated into part of ndergradate cheical, echanical and petrole engineering crricla. A orked eaple is inclded to sho the tility of the approach. Keyords Heat transfer coefficient, Nsselt nber, isotheral flat plate. Noenclatre heat transfer area [ ] average heat transfer coefficient [ ( )] local heat transfer coefficient at [ ( )] flid theral condctivity [ ( )] length of the plate [ ] average Nsselt nber correlation [ ] flid Prandtl nber [ ] total heat transfer rate [ ] Reynolds nber [ ] freestrea teperatre [ o or ] constant srface teperatre [ o or ] freestrea velocity [ ] aial coordinate easre fro the leading edge [ ] critical aial coordinate here lainar flo transitions to trblent flo [ ] transverse coordinate [ ] idth of the plate [ ] flid dynaic viscosity [ ] flid density [ ] Sbscripts la ied s trb average or at = fro the leading edge lainar ied srface trblent local at fro the leading edge freestrea

2 Introdction Shear stress, heat transfer and/or ass transfer for parallel flo over a flat plate is an iportant part of an ndergradate cheical engineering, echanical engineering or petrole engineering crricl. For ease of discssion and ithot loss of generality, e shall present or discssion sing heat transfer. he sae approach can be applied to evalate shear stress or ass transfer for the sae flo arrangeent. Heat transfer fro parallel flo over a flat plate fors an integral part of heat transfer coverage and is detailed in standard heat transfer 1, 2 or theral flids 3, 4 tets. Heat transfer rate fro a flat plate is, aongst other things, a fnction of the average Nsselt nber. he average Nsselt nber ith lainar bondary layer can be obtained analytically. here is no eact soltion for trblent bondary layer. he average Nsselt nber for trblent bondary layer is dedced fro eperiental observations of the skin friction coefficient and the odified Reynolds analogy. Undergradate tets then proceed to evalate the average Nsselt nber for ied bondary layer conditions by integrating these to average Nsselt nbers. In this paper, e present an alternative approach here the Nsselt nber for ied bondary layer conditions is obtained sing the average Nsselt nber for lainar bondary layer and the average Nsselt nber for trblent bondary ithot the foral integration of the heat transfer coefficients for the to types of bondary layers. his approach is pictorial and can be sed to evalate the average skin friction and Sherood nber for ied bondary layer conditions for parallel flo over a constant vale (zero velocity for skin friction and constant concentration for Sherood nber) at the flat plate. he reainder of this paper is divided into for sections. he physical sitation considered in this paper is discssed in the net section. he evalations of the Nsselt nbers for different types of bondary layers are then presented. Or approach to evalate the Nsselt nber for ied bondary layer conditions is then eplained. We then deonstrate or approach sing an advanced proble. Soe conclding rearks are given to conclde the paper. Physical sitation In this article, e shall asse that the Netonian flid ith constant properties. he flo is steady ith negligible viscos dissipation and pressre gradient. Figre 1 shos the physical sitation considered in this article. A flid flos ith a freestrea velocity and a freestrea teperatre parallel to an isotheral flat plate aintained at a constant srface teperatre. he length of the plate is and the idth of the plate is. here are three possible scenarios to consider hen analysing the average heat transfer fro a flat plate. In the first sitation shon in Fig. 1a, a lainar bondary layer develops starting at the leading edge ( ). If the plate is short sch that the local Reynolds nber at the trailing edge of the plate ( ) is less than the critical Reynold nber for the flo to transition into trblent flo, the lainar bondary layer spans the hole length of the plate. If the length of the plate is sfficiently long, the flo transitions to trblent flo at hen the critical Reynolds nber is reached as depicted in Fig. 1b. Here, e asse the flo transition abrptly fro lainar into trblent flo neglecting the transitional flo region as shon in Fig. 1b. o avoid confsion, the transition region ill not be shon in sbseqent figres. Figre 1c shos a sitation here the flo reaches the leading edge distrbed. As a reslt, trblent bondary layer starts to for and persists over the hole plate. For sae flo conditions,. here is hoever, no relation beteen and the other to lengths.

3 ainar ainar ransition rblent rblent y s y s y s la c trb ied (a) (b) (c) Figre 1. op and side vies of the physical sitation considered (a) lainar bondary layer, (b) ied bondary layer and (c) trblent bondary layer. he total heat transfer rate Using Neton s la of cooling, the total heat transfer rate to-or-fro a srface is ( ) (1) here is the total heat transfer rate, is the average heat transfer coefficient and is the heat transfer area. he average heat transfer coefficient for flo over a flat plate can be calclated fro the Nsselt nber correlations. hese to paraeters are related throgh (2) here is the flid theral condctivity. Using Eq. (2) and the area for a rectanglar flat plate in Eq. (1) gives ( ) (3) he total heat transfer rate fro a flat plate can then be evalated sing Eq. (3) once the average Nsselt nber is obtained. his is described net. Average heat transfer coefficient and average Nsselt nber he Nsselt nber correlations needed in Eq. (3) are obtained fro the heat transfer coefficients. For ease of discssion and ithot loss of generality, let s call the local heat transfer coefficient at any aial location as. he average heat transfer coefficient over the hole length of the plate can then be calclated throgh (4)

4 he average Nsselt nber is then obtained sing Eq. (2). he average Nsselt nbers for the lainar bondary layer and for trblent bondary layer are described net. he coon practice to evalate the average Nsselt nber for ied bondary layer is then discssed. We shall then describe an approach to obtain the average Nsselt nber for ied bondary layer sing the average Nsselt nbers for lainar and trblent bondary layers ithot eplicitly perforing the integration over the to bondary layers. ainar bondary layer When a lainar bondary layer spans the hole length of an isotheral plate, the local heat transfer coefficient for a flid ith Prandtl nber can be obtained analytically as (5) here is the local Reynolds nber defined as (6) In Eq. (6), is the flid density and is the flid dynaic viscosity. he average Nsselt nber over the length of the plate is obtained sing Eqs. (2) and (4) to (6) as (7) here is the Reynolds nber defined as 5, 6 (8) rblent bondary layer As a reslt of the inherent nsteady natre of trblent flo, there is no eact soltion for trblent bondary layer. Fro eperiental observations, the local heat transfer coefficient for parallel flo over an isotheral flat plate here trblent bondary layer prevails fro the leading edge is given by (9) Using the sae approach as above, the average Nsselt nber can be obtained as (10) Eqation (10) is valid hen and. 1 Mied bondary layer Coon approach For the sitation shon in Fig. 1b here a lainar bondary layer begins to develop at the leading edge and the length of the plate is sfficiently long for the flo to transition into trblent flo at hen the critical Reynolds nber is reached, ied lainar and

5 trblent bondary layers are encontered. For sch sitation, the average heat transfer coefficient is obtained fro the lainar bondary layer and the trblent bondary layer heat transfer coefficients as ( ) (11) Using Eqs. (5), (9) and (11), the average Nsselt nber for ied bondary layer conditions here the lainar bondary layer transitions into a trblent bondary layer at a critical Reynolds nber of becoes ( ) (12) In Eq. (12), the critical Reynolds nber here the flo transitions fro lainar to trblent is (13) When the flo transitions at, Eq. (12) redces to the coonly knon average Nsselt nber correlation for the ied bondary layer conditions of ( ) (14) Rearks In this approach, the average heat transfer coefficient for ied bondary layer conditions is evalated by cobining the average lainar heat transfer coefficient and the average trblent heat transfer coefficient. Stdents are orking ith three sets of eqations naely, one set for lainar bondary layer, a second set for trblent bondary layer and a third set for ied bondary layers. Once the ied bondary layer conditions correlation (Eq. 12) is obtained, soe stdents treat this as a forla and copte the anser ithot nderstanding the physics. Another approach is described net. Or approach We present an approach to evalate the average Nsselt nber for a ied bondary conditions aking se of the eisting correlations for lainar bondary layer and trblent bondary layer ithot carrying ot the atheatical evalation of Eq. (11). Figre 2 shos the top vie of a flat plate ith ied bondary conditions fro hich the total heat transfer rate fro the entire srface st be evalated. As indicated in Eq. (3), this can be done once e kno the average Nsselt nber for the hole plate. We propose to evalate the average Nsselt nber for the ied bondary conditions sing the average Nsselt nber for lainar bondary layer (Eq. 7) and the average Nsselt nber for trblent bondary layer (Eq. 10) as shon in Fig. 2. Using Fig. 2, the average Nsselt nber for ied bondary conditions is evalated as (15)

6 ainar rblent rblent rblent ainar c c c Figre 2. op vies of the physical sitation considered. Sbstitting Eqs. (7) and (10) ith the appropriate lengths, Eq. (15) becoes (16) Eqation (16) can be readily siplified to ( ) (17) Coparing Eq. (17) and Eq. (12), it is clear that or approach prodces the sae average Nsselt nber ith the coon approach. his is done ithot the foral atheatical treatent of Eq. (11). Eaple In this proble, e deonstrate the se of or approach to a ore advanced proble. We also discss the advantage of this approach over the crrent practice. Proble stateent Air at atospheric pressre flos parallel over a 3- long and 1- ide flat plate. he freestrea air teperatre is hile the freestrea air velocity is. he plate srface teperatre is kept at. Assing lainar bondary layer begins to develop at the leading edge of the flat plate and transitions into trblent bondary layer at, hat is the total cooling rate beteen and of the plate hen the plate is aintained at the specified srface teperatre? Air properties at the fil teperatre are, ( ),,. Note that the standard overall Nsselt nber correlation shon in Eq. (14) cannot be sed as. Scheatic

7 ainar (a) rblent ainar rblent (b) Figre 3. op vies of the heat transfer rate beteen 1.5 bondary layer type(s) and (b) inclding bondary layer types. Analysis Fro Fig. 3a, e can evalate the total heat transfer rate beteen (a) ithot sing (18) Using Neton s la of cooling, Eq. (18) can be frther epanded as ( ) ( ) (19) We shall no focs on the evalations of the average Nsselt nbers in Eq. (19). he net step is to find ot hat type(s) of bondary layer(s) are eperienced. o do this, e shall find the location here the critical Reynolds nber is reached. Using the definition of the Reynolds nber (Eq. 13), e have (20) As the location here the flo transition fro lainar to trblent is, ied bondary layer conditions are encontered in the evalations of and only lainar bondary layer prevails in the evalation of. his is depicted in Fig. 3b. he ied bondary layers can be obtained ith the approach otlined in Fig. 2 as ( ) (21)

8 he Reynolds nber is (22) he Nsselt nber then becoes [ ( ) ( ) ( ) ] ( ) or (23) he lainar bondary layer is obtained fro Eq. (7) as (24) Fro Eq. (24), the Reynolds nber is (25) he is then ( ) ( ) (26) Using Eqs. (23) and (26) in Eq. (19), the total heat transfer rate is ( ) ( ) ( ) (27) Rearks In this eaple, e shoed ho the proposed approach can be sed in solving a ore advanced proble. he pictorial approach shon in Fig. 3a is a direct etension of or approach to calclate ied bondary layer conditions in Fig. 2. Using this approach, different cobinations of bondary layer types can be handled ithot the need to forally evalate the Nsselt nber throgh integrations. Initially, the stdents can focs on the physics of the proble and not be concerned ith the foral atheatics. Once, the physics have been properly nderstood, the foral atheatics can then be introdced. Conclding rearks We present an alternative approach to evalate the average Nsselt nber for parallel flo over a flat plate aintained at constant (zero all velocity, nifor srface teperatre and nifor srface concentration) vales. he approach is deonstrated sing the total heat transfer rate. Applications of the approach to evalate skin friction and ass transfer rate do not reqire ne concepts and are straight forard. hs, they are left to the eplorations of interested readers.

9 Declaration of Conflicting Interests he athors declared no potential conflicts of interest ith respect to the research, athorship, and/or pblication of this article. Fnding he athors received no financial spport for the research, athorship, and/or pblication of this article. References 1. Bergan, avine A, Incropera F, et al. Fndaentals of Heat and Mass ransfer. John Wiley & Sons, Inc., Cengel Y and Ghajar A. Heat and Mass ransfer: Fndaentals and Applications. 5 ed.: Mc Gra Hill, Cengel Y, rner R and Cibala J. Fndaentals of heral-flid Sciences. 5 ed.: Mc Gra Hill, rns S. heral-flid Sciences: An Integrated Approach. Cabridge University Press, Holland F and Bragg R. Flid Flo for Cheical Engineers. Btterorth-Heineann, 1995, p Darby R and Chhabra R. Cheical Engineering Flid Mechanics. CRC Press, 2016, p.577.