Calculation of the temperature of boundary layer beside wall with time-dependent heat transfer coefficient

Transcription

1 Ŕ periodica polytechnica Mechanical Engineering 54/ doi: /pp.me web: me c Periodica Polytechnica 21 RESERCH RTICLE Calculation of the temperature of boundary layer beide wall with time-dependent heat tranfer coefficient ttila Fodor Received btract Thi paper propoe to invetigate the change in the temperature of external wall boundary layer of building when the heat tranfer coefficient reache it tationary tate in time exponentially. We eek the olution to the one-dimenional parabolic partial differential equation decribing the heat tranfer proce under pecial boundary condition. The earch for the olution originate from the olution of a Volterra integral equation of the econd kind. The kernel of the Volterra integral equation i lightly ingular therefore it olution i calculated numerically by one of the mot efficient collocation metho. Uing the Euler approach an iterative calculation algorithm i obtained, to be implemented through a programme written in the Maple computer algebra ytem. Change in the temperature of the external boundary of brick wall and wall inulated with polytyrene foam are calculated. The concluion i reached that the external temperature of the inulated wall matche the air temperature ooner than that of the brick wall. Keywor heat conduction lightly ingular Volterra integral equation numerical olution heat tranfer coefficient relaxation time ttila Fodor BME, H-1521 Budapet, Hungary fodor@pmmk.pte.hu 1 Introduction The heat tranfer and thermal conductivity of the wall boundary layer of building enure the temperature potential inide and outide the wall boundarie of building. The majority of the relevant model focu on temperature change at internal point in the wall boundary depending on the variou inulation layer. It i aumed that the temperature of the external wall boundary and the average temperature of the outide air urrounding the wall are the ame i.e. the outide temperature become contant and the temperature of the internal wall boundary i alo contant. Thi problem i not dicued by thi paper. We invetigate how quickly the external wall boundary adopt the temperature of the air boundary layer under a contant temperature of the internal wall boundary. Thi only require the examination of the implified model of the wall boundary preented in Fig. 1. In the mathematical model the wall i regarded a one of infinite thickne. It temperature i denoted by ux, t at poition x > and time t >. It i ufficient to deal with the one dimenional cae a it can be aumed that along the length of the wall the temperature i ditributed equally at each vertical croection. In accordance with the denotation u, t denote the temperature of the external boundary wall layer. The external layer of the wall i in contact with the ambient air. The internal layer of the wall mixe with the air of the room, aumed to have a contant temperature. The wall reit the incoming heat flow perpendicular to the wall therefore in the "air -wall" boundary layer the temperature of the air i different than the urface temperature of the wall. The temperature of the external wall boundary take ome time to adopt the average temperature of the outdoor air. denote the average temperature of the air flow. In winter it i typically < µ, t i.e. the outdoor air cool the wall. In ummer, t <, i.e. the outdoor air warm the wall. The model can be applied to both cae but implified criteria u, = i ued. The initial temperature t = ec of the wall layer i thu C and <, i.e. the air flow gradually warm the external wall boundary. Then, if t ten to infinity the temperature of the external wall boundary reache air tem- Time-dependent heat tranfer coefficient

2 Fig. 1. Simplified model of wall boundary Fig. 2. Curve of the α heat tranfer coefficient in the boundary layer with relaxation time T= 2, 5, 1, 15 ec downwar from top perature. Thi change will be calculated numerically. The thermal impact of the air on the wall i characterized by convective heat tranfer coefficient α. It i aumed that the thermal tranmittance coefficient αt of the wall change in time αt = α 1 e t T 1 exponentially [1] where α i the contant convective heat tranfer coefficient of the external ide, α = 23 W/m 2 K and T > i the o-called relaxation time with a dimenion inec. The change of the heat tranfer coefficient in time ipreented in Fig. 2. In Fig. 2 the graph how the change of the convective heat tranfercoefficient for relaxation time T = 2, 5, 1 and 15. T increae the graph go downwar i.e. the function reache a tate of balance later if T i higher. Time t required to reach the ame degree of balance i in direct proportion to the length of T. For example time t required to reach,95 time balance α i attained by olving equation, 95 = 1 e t T where the olution i t = T. Our objective i to determine by continual approache function u, t which how in time the change of temperature of the external wall boundary layer x =, taking into account that the air flow parallel to the wall modifie in time the αt heat tranfer factor of the wall-air boundary layer according to exponential equation preented in formula 1. The ubject of thi paper therefore belong to the topic of boundary layer theory. The partial equation i written up and will be olved uing the Volterra equation. In ection 2 of thi paper the partial differential equation determining temperature ux, t of the wall will be preented together with the neceary boundary condition. To eek the olution the Laplace tranformation of the olution in time t will be ued. To attain the reult we need to olve a Volterra integral equation of the econd kind. Section 3 will preent the formula determining the iterative olution of the obtained Volterra integral equation. Uing thee formula a Maple computer algebra programme will be written calculating the approaching value of the olution in an iterative way. Iue relating to the numerical preciion and tability of the approaching value can be conulted in the bibliography ee [1, 3] and [4]. Section 4 deal with two type of wall: brick wall and wall inulated with polytyrene. The neceary parameter are calculated for thee two cae and the numerical olution are written up. Finally a concluion will be drawn about the behaviour of the boundary layer of the two type of wall. 2 The mathematical model of the problem and deduction of the Volterra integral equation 2.1 Governing equation and olution procedure Temperature ux, t of the wall atifie the regular heat conduction equation 2 u x, t = a u x, t t x2 with the following boundary condition: u x, = u, t = x > ; t > 2.a 2.b 2.c 16 Per. Pol. Mech. Eng. ttila Fodor

3 t λ x x= u x, = α 1 e T t u, t 2.d where: λ i the thermal conductivity of the wall and the inulation material, W/mK, α i the tationary convective heat tranfer coefficient, W/m 2 K, a = ρc λ thermal diffuivity, m2 / if the temperature i not evenly ditributed it define the peed of equalization ρ denity, kg/m 3, c pecific heat capacity, J/kg K. Dynamic equation 2.d play a crucial role in the proce. The peed of the change in temperature perpendicularly impacting the wall i proportional to the difference in the temperature of the wall urface and the air and the proportion factor i heat tranfer coefficient α obtained by formula Step of deducing the Volterra integral equation Temperature u, t of the external wall boundary layer i obtained from Eq. 2.a-d and it i not neceary to calculate temperature ux, t x > of the internal point of the wall. We need to deduce from Eq. 2.a-d a o-called Volterra integral equation only containing function u, t, which ha a lightly ingular kernel of the econd kind. From the theory of Volterra integral equation of the econd kind the method of numerical olution i ued where the Euler approach i applied to calculate the value of the integral. Thi cheme i the implet verion of the collocation method. The integral equation to calculate temperature u, t of the wall-air boundary layer i derived through the following tep. working of thi problem Garbai [2] too. Step 1 Variable t, x and ux, t in Eq. 2.a-d are replaced tranformation X = x ; τ = t u x, t ; U X, t = T a T by new dimenionle variable X, τ and dependent variable UX,τ. Thee variable turn Eq. 2.a-d into the following impler verion: U X, τ = 2 t X U X, τ x= where = α at λ. U X, τ X > ;τ > 3.a X 2 U X, = U, τ = 3.b 3.c = 1 e τ U, τ 1 3.d In the following ection we will deal with the olution to Eq. 3.a-d but variable X, τ and UX, τ are replaced by the original variable x, t and ux, t. in Eq. 3.a-d boundary layer U,τ tarted from the initial τ = C and following the tranformation the mean air temperature went up to 1 C we will invetigate the curve along which the function of temperature U,τ increae from to 1. If we wih to return to the original temperature [ C] the calculated U, τ value nee to be multiplied by becaue u x, t = U X, τ fter the tranformation the variable value of τ how how many time it excee relaxation time T. U, τ = r mean the temperature that the boundary layer adopt at point of time t = r T, i.e. r time of relaxation time T. Step 2 To find olution ux,t to parabolic partial differential equation 3.a-d the Laplace tranformation i applied to variable t. The Laplace tranform of wx, for t of function x,t i defined by formula L t u x, t = e t u x, t dt = w x, The Laplace tranform of partial differential Eq. 3.a turn into L t t u x, t 2 = L t u x, t x2 w x, u x, = 2 w x, x2 w x, = 2 w x, 4 x2 ordinary differential equation where the uniformity for the derived Laplace tranform and initial condition 3.b ux, = are ued. Step 3 With regard to variable x Eq. 4 i an ordinary linear differential equation of the econd kind with a contant coefficient where i one of the parameter. It general olution i w x, = F1 e x + F2 e x where F1 and F2 are the dicretionary function of parameter. in the cae of x ux, t i zero according to 3.c therefore lim w x, = alo applie to the Laplace tranform of x wx, for t. Thi will only be atified if F1= becaue the wall i therefore w x, = F2 e x 5 Step 4 We derive the Laplace tranform of wx, from 5 by x and re-ubtitute wx, from Eq. 5 into the obtained equation x w x, = F2 e x x w x, = w x, 6 Time-dependent heat tranfer coefficient

4 Step 5 Since we are intereted in the temperature of the boundary layer ubtitution x= nee be performed in Eq. 6. x x= w x, = w, t i 1 e g = 1 + π i 1 j= t j+1 1 e g where F = L t f t and and H = L t h t. fter arranging the equation Step 6 If the Laplace tranformation i ued in variable t for boundaryeq. 3.d g t = 1 t 1 e g t 13 π x x= w x, = L t 1 e t u, t 8 i obtained where the Laplace tranformation equality wa ued L t 1 e t = = the left ide of Eq. 7 and 8 are the ame the right ide are alo equal w, = L t 1 e t u, t Both ide of the equation hould be divided by expreion and it hould be taken into account that w, = L t u, t L t u, t = L t 1 e t u, t 1 Step 7 Eq. 1 include the Laplace tranform L t u, t of temperature function u, t of the wall boundary layer and the Laplace tranform of function 1 e t u, t. In function Eq can be brought to zero if u, t i ubtituted by u, t = g t + 1 t 11 Baed on the linearity of Laplace tranformation and correlation 9 L t g t + 1 = 1 + L t 1 e t g t + 1 L t g t + 1 = L t 1 e t g t 12 Step 8 The invere Laplace tranformation hould be ued for both ide of Eq. 12 and the invere Laplace tranformation corre- lation L t 1 = 1 t hould be applied a well a the convolution formula written up with invere g t i = Volterra integral equation of the econd kind i obtained in which kernel K t, = 1 t i not interpreted at the = t end point of the integral interval. 3 Numerical olution to Volterra integral equation uing the Euler approach We have een that function g t = u, t 1 atifie the lightly ingular Volterra integral equation of the econd kind 13 where function ux, t i the olution to group of Eq. 3.a- 3.d. Let ee < t <H time interval and it ditribution to N>1 equal part B N t = < t 1 < t 2 < t 3 < < t i < t i+1 <... < t N = H where t i = iδ i the i-th diviion point and δ = H N i the pacing i =, 1, 2,..., N. We eek the approaching value g t 1, g t 2,..., g t N to olution gt of the Volterra integral Eq.13 at B N diviion point. We know the initial g=-1 value. It i aumed that approaching value g t 1, g t 2,..., g t i 1, are known. To obtain the value of g t i we need to break up the integral from the integral equation into the um of integral from partial interval B N g t i = t i i 1 j= 1 e g = π t j+1 1 e g 14 1 where it may be: i = 1, 2, 3,..., N. In integral from [, +1 ] partial interval multiplying factor f = 1 e g hould be approached by contant value f recorded in the left end point = while ingular kernel hould be integrated by in the given integral. Thi give u the following +1 1 e g = 1 e g +1 ti 1 e g 2 ti t i +1 1 ti = Per. Pol. Mech. Eng. ttila Fodor

5 for all j =, 1, 2,..., i 1. Therefore to obtain the approaching value for gt i the following recurive calculation formula i ued g t i = i 1 j= 1 e g ti t i On the right ide of recurive formula 16 value g t = 1, g t 1, g t 2,..., g t i 1 are preented therefore the value of function g can be calculated tep by tep. To calculate gt 2 i only neceary to know g t = 1. To calculate g t 2 the previouly calculated g t 1 i alo required. To calculate g t 3 both g t 1 and g t 2 previouly calculated are needed. The calculation of the next temperature figure alway ue the previouly obtained value of temperature. 4 Numerical reult for different wall urface There i a ingle parameter in Volterra integral Eq. 13 which can be calculated uing correlation = α at λ with the earlier mentioned data of the urface material of the wall. In the cae of a B3 brick wall λ=.647, ρ=146 and c=.88. If relaxation time T = 1 ec i ued in the calculation then =.7977 i obtained. For a B3 brick wall relaxation time T = 2 ec = i obtained. For inulating material polytyrene foam the ame data are λ=.47 with parameter ρ=15, c=1.46. If T = 1 then = while for T = 2 the parameter i = It can be thu tated that the value of coefficient in the Volterra integral Eq. 13 increae if better inulting material are ued relaxation time T i increaed. The following numerical and graphical reult were attained through the Maple computer algebra ytem. Maple function wa written for recurive correlation 16. Uing element g =-1, g 1, g 2,..., g i 1 in a lit it calculate the next g i element through formula Fig. 3. Surface temperature function u, t in the cae of B3 brick wall 1 C outdoor air temperature of C to 1 C outdoor air temperature =1 C. The horizontal line i the t-time axi meaured in ec and the vertical line i wall urface temperature u,t meaured in C. In Fig. 4 the graph for function u, t ha alo been drawn in a coordinate ytem where logarithm formula x = log 1 2 T t wa ued on the horizontal axi. Thi figure illutrate better the change of the temperature function for mall t time. Fig. 4. Function u,t with parameter =.8 and horizontal cale x = log 1 2 T t The approaching calculation are repeated for parameter correponding to =22 polytyrene inulation. The reult i preented in Fig. 5. iteration : = i 1 i e jδ g j i j δ i j 1 δ j= 17 Chooing parameter =.8 which correpon to brick wall B3 with relaxation time T = 1 [ec] the graph how the approaching value for urface temperature u, t = gt + 1. The Fig. 3 preent the curve along which the external layer of the brick wall increae from the initial temperature of C to ued Fig. 5. Surface temperature function u,t if polytyrene foam =22 i The graph for function u, t with parameter value =22 wa written up according to the Fig. 6 where the horizontal axi Time-dependent heat tranfer coefficient

6 i logarithmically caled. Fig. 6. Function u,t with parameter =22 and horizontal cale x = log 1 2 T t Drawing up the graph in Fig. 3 and 5 in a common coordinate ytem the difference between the two graph i highly viible a hown in Fig. 7. The higher graph how how the external wall boundary layer adapt to air temperature =1 C in the cae of a B3 brick wall with polytyrene inulation and the lower graph how the curve for a wall without inulation. Fig. 8. Surface temperature u, t hown on a logarithmic cale The logarithmic cale ued on the horizontal axi revere the place of the two graph compared to Fig. 7. In Fig. 8 the left graph applie to the B3 brick wall while the right graph applie to the polytyrene inulation. wall Fig. 7. Surface temperature u,t for inulated and uninulated B3 brick Fig. 6 how that the temperature of the external urface of the inulated wall almot immediately adopt the temperature of the outdoor air while thi proce take longer for an uninulated brick wall. The explanation for thi fact i that inulation doe not carry the outdoor temperature inide but form an obtacle to the preading of the heat o there i no equalization between the temperature of the external and internal layer in term of thermal tranmittance. The brick wall, however, tranmit the external volume of heat better and a kind of equalization proce tart between the indoor and outdoor temperature. a reult the temperature of the external urface of the B3 brick wall i impacted by indoor temperature C i.e. the brick wall carrie the volume of heat inide and reache outdoor temperature 1 C later than the external urface of the inulated wall. The graph of the temperature function preented in Fig. 4 and 6 were put into the ame coordinate ytem Fig. 8 where the horizontal axi i logarithmically caled. 5 Concluion Baed on Fig. 7 the following concluion can be drawn about the adaptation capability of the external wall urface: the temperature of the wall urface adapt more quickly in time to the air temperature if the wall i inulated. Thi phenomenon i explained by the fact that brick in an uninulated wall carry the heat inide and thu the temperature of the external wall urface adapt more lowly to the air temperature. In thi cae the indoor temperature influence the outdoor urface temperature. If the wall i inulated, however, the indoor temperature cannot impact the temperature of the external wall urface to the ame extent a the inulation provide a thermal barrier between the external and internal temperature and adopt the outdoor air temperature almot immediately. Reference 1 Becker N M, Bivin R L, Hu Y C, Murphy H D, Jr.White B, Wing G.M, Heat diffuion with time-dependent convective boundary condition, International Journal for numerical metho in energinering , Garbai L, Fénye T, The newet reult of heat conduction theory, Periodica polytechnica, Ser. Mech. Eng. 44 2, no. 2, Diogo T, Lima P, Rebelo M, Computational metho for a nonlinear Volterra integral equation, Proceeding of Hercma 25, Ori P, Product Integration for Volterra integral equation of the econd kind with weakly ingular kernel, Mathematic of Computation , no. 215, Per. Pol. Mech. Eng. ttila Fodor