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23-Chem-B1 Transport Phenomena · December 2017

Question 4 of 6: B2 — Transient heating of a well-stirred tank by a coil

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. EGBC 16-CHEM-B1 Transport Phenomena, December 2017, 3 hours, open-book (one textbook permitted, no loose notes). Six problems in three sections (A Fluid Mechanics, B Heat Transfer, C Mass Transfer), each worth 25 marks; candidates attempt one problem from each section plus a fourth from any section, and only the first four in the answer book are marked. All six problems are solved below as a complete study resource. Appendix A of the paper supplies the equations of change (continuity, Navier–Stokes, energy and species-continuity tables) in rectangular, cylindrical and spherical coordinates; these are quoted rather than re-derived.

Reference texts: R. B. Bird, W. E. Stewart & E. N. Lightfoot, Transport Phenomena (2nd ed., Wiley) — the equations of change and the differential momentum / energy / species balances; J. R. Welty, C. E. Wicks, R. E. Wilson & G. L. Rorrer, Fundamentals of Momentum, Heat and Mass Transfer (Wiley) — pipe friction, the transfer analogies, boundary-layer and film coefficients; F. P. Incropera & D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley) — internal-flow convection and the sphere/flat-plate correlations; F. M. White, Fluid Mechanics (McGraw-Hill) — the three-reservoir branching-pipe problem; R. H. Perry & D. W. Green, Perry’s Chemical Engineers’ Handbook — transport properties.

Question 4: B2 — Transient heating of a well-stirred tank by a coil (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A well-stirred (perfectly mixed) tank holding a mass $M$ of liquid with specific heat $c$, initially at $T_{in}$. Hot oil (specific heat $c_{p,oil}$) flows through an immersed coil at mass rate $\dot m$. The coil temperature is $T_o$, which is taken as the temperature at which the oil enters the coil. The paper’s assumption that “the exit temperature is the same as that inside the tank” means the oil leaves the coil at the tank temperature $T(t)$. Heat losses to the surroundings and the heat capacity of the coil are neglected. $M$, $c$ and $c_{p,oil}$ are not printed and stay as symbols.

Find. (a) $T(t)$ for the tank contents; (b) the limit $T(t\to\infty)$.

liquid surface T(t) — well mixed mass M, specific heat c hot oil ṁ: in at T_o, out at T ↻
Fig. B2: A perfectly mixed tank heated by hot oil in an internal coil. The oil enters at $T_o$ and, by the stated assumption, leaves at the tank temperature $T(t)$. It therefore gives up $\dot m c_{p,oil}(T_o-T)$, and the tank temperature relaxes exponentially from $T_{in}$ toward $T_o$.

Approach. This is an unsteady macroscopic energy balance. Perfect mixing gives the tank one temperature $T(t)$. An energy balance on the oil stream gives the heat delivered: the oil enters at $T_o$ and leaves at $T$, so the coil supplies $\dot m c_{p,oil}(T_o-T)$. The resulting first-order linear ODE integrates directly, and letting $t\to\infty$ gives part (b).

  1. (a) Heat delivered by the oil. A steady energy balance on the oil flowing through the coil (its hold-up is negligible) gives $$\dot Q=\dot m\,c_{p,oil}\,(T_{oil,in}-T_{oil,out})=\dot m\,c_{p,oil}\,(T_o-T),$$ because the oil enters at the coil temperature $T_o$ and, per the paper’s assumption, leaves at the tank temperature $T$.
  2. (a) Unsteady energy balance on the tank. The tank’s energy accumulates at the rate that heat arrives from the coil: $$Mc\,\frac{dT}{dt}=\dot m\,c_{p,oil}\,(T_o-T),\qquad T(0)=T_{in}.$$ The well-stirred assumption is what lets a single $T(t)$ describe the whole tank.
  3. Separate and integrate. Let $\theta=T_o-T$, so $d\theta/dt=-dT/dt$ and $$\frac{d\theta}{\theta}=-\frac{\dot m\,c_{p,oil}}{Mc}\,dt\ \Longrightarrow\ \ln\frac{\theta}{\theta_0}=-\frac{\dot m\,c_{p,oil}}{Mc}\,t,\qquad \theta_0=T_o-T_{in}.$$
  4. Temperature profile in time. Restoring $\theta=T_o-T$, $$\boxed{\,T(t)=T_o-\big(T_o-T_{in}\big)\exp\!\Big(-\frac{\dot m\,c_{p,oil}}{Mc}\,t\Big).}$$ The group $\tau=Mc/(\dot m c_{p,oil})$ is the thermal time constant: after $t=\tau$ the tank has closed $1-e^{-1}\approx63\%$ of the gap to $T_o$.
  5. (b) Long-time limit. As $t\to\infty$ the exponential vanishes, so $$\boxed{\,T(\infty)=T_o.}$$ The tank approaches the temperature of the oil entering the coil. It can never exceed that temperature, because the oil is its only heat source.
ResultExpression
(a) Tank temperature vs. time$T(t)=T_o-(T_o-T_{in})\,e^{-\dot m c_{p,oil}t/Mc}$
Thermal time constant$\tau=Mc/(\dot m\,c_{p,oil})$
(b) Steady limit$T(\infty)=T_o$
Check — what the “exit temperature” assumption does

For a coil with finite conductance $UA$, the oil leaves at $T+(T_o-T)e^{-UA/\dot m c_{p,oil}}$, so the coil delivers $\dot Q=\varepsilon\,\dot m c_{p,oil}(T_o-T)$ with $\varepsilon=1-e^{-UA/\dot m c_{p,oil}}$. The paper’s assumption that the oil exits at the tank temperature is the limit $\varepsilon=1$, where $\dot m c_{p,oil}$ sets the time constant. In the opposite limit (very high oil flow, coil surface uniformly at $T_o$), $\dot m c_{p,oil}$ is replaced by $UA$. Either way the result has the same exponential form, and $T(\infty)=T_o$. $U$ and $A$ are not given, so the printed assumption is the one used.