Int J ttcat $lavr lrutt*ti'r '~,~1 ~3. X,v, q pp 2051-2053. 1990 (X)I ~ ~MO q()$3 00+O(X) Pnnted :n Great Bream ~ 19'~1) Pergamon Press r, lc TECHNICAL NOTES Combined thermal-momentum start-up in long pipes GREGORY S. PATIENCE~ and ANIL K. MEHROTRA Department of Chemical and Petroleum Engineering, The University of Calgary, Calgary, Alberta, Canada T2N IN4 (Received 13 June 1989 and in final form 20 Nocember 1989) 1. INTRODUCTION THE FLOW regulation of fluid systems involving transient laminar forced convection has become important in con- nection with high performance heat transfer equipment requiring precision control. Previous mathematical and experimental investigations have been limited mostly to the cases of steady flow with changes in wall heat flux, inlet temperature or wall temperature [1-71 . The problem con- cerning combined thermal and momentum start-up has been given little attention. Creffand Andre [8] provided numerical solutions for certain flow conditions including a pulsating flow situation. In the combined thermal-momentum start-up problem, both the velocity and temperature profiles change with time. The solution of the combined thermal-momentum problem is complicated mathematically due to coupling of the partial differential equations of energy, momentum and continuity. In the approach used here for the constant-property combined thermal-momentum start-up in long pipes, the momentum equation is decoupled from the energy equation. Szymanski's [9] analytical velocity profile for the momentum equation is substituted into the energy equation which is solved numerically for the combined thermal-momentum start-up. The results for the variation of Nusselt number and average temperature are presented at various dimensionless times. A thermal start-up time parameter (Fo0.99)--defined in terms of Fourier number at which the average fluid tem- perature reaches 99% of the corresponding steady state value--is introduced to qualify the transient thermal flow development along the pipe length. Finally, the numerical results are correlated to provide an estimation procedure for the start-up time. 2. MATHEMATICAL FORMULATION AND SOLUTION PROCEDURE Consider a long circular pipe of radius R that is filled initially (i.e. t = 0) with a constant-property, incompressible, Newtonian fluid at a constant temperature To throughout. At t > 0, the fluid is exposed to a constant pressure gradient (-APL) in the axial direction while a step change in tem- perature is simultaneously imposed at the wall (r = R, 0 < .-). Note that there are no entrance effects for flow development in long pipes, hence u =- u(r, t). The equations governing the heat and momentum transfer in axisymmetric transient laminar flow of a constant- property, incompressible, Newtonian fluid in a long hori- zontal straight pipe are t Present address: Ecole Polytechnique, Montreal, Quebec, Canada H3C 3A7. pC ~i T(r,z.t)+u{r,t) ~/zT(r,c,t) r cr cr T(r,c,t) (1) p: u(r.l) +- ~-|r\-u(r,t)|. (2) ~r = ~ rcr L ,, The initial conditions for the combined thermal momentunl start-up problem are T(r,z,O)= To; 0<z, 0<r< R (3) u(r,O)=O; 0<r<R. (4) The boundary conditions are u(R.t)=O: 0<t (5) : u(0, t)=0: 0<t (6) er T(r,O,t)= To; 0<r<R, 0<t (7) T(R,z,t)= T~: O<z. 0<t (8) .-T(O,-.t)=O: 0<--. ()<t. (9) cr Since the fluid properties are assumed to be independent of temperature, the momentum equation is not coupled to the energy equation and therefore may' be solved analytically [4]. The following velocity distribution for flow start-up in long pipes was given by Szymanski [9] q5 =(I-~Z)-8 J0 (:~,-,) ~7-,, --c o (I0) % :~; J. (:~,) where :t, are the positive roots of J0(:t,). The terms 4), ,~ and in equation (I0) are u r l~l c~= -APRZ 41~L" ~-R" r-pR'- (li) Note that the pressure gradient (-AP/L) should not be greater than 8000 I*"/pR 3 in order for the flow to be laminar (i.e. Re < 2000). The use of equation (10) as the solution to equation (2) simplifies the combined thermal momentum start-up problem to a single partial differential equation (i.e. equation ('1)). Equation (1) was expressed in finite difference form and the resulting set of non-linear equations was solved numerically using a Newton-Raphson convergence scheme. The details of grid distribution and numerical technique are presented elsewhere [I0, 11]. 3. RESULTS AND DISCUSSION For comparison, two steady-flow thermal-entry problems involving a step change in wall temperature are considered. 2051