Heat and Mass Transfer (CHE210H1S) Winter term 2015 Created by: Prof. Passeport

Last Updated: Apr. 6th, 2015 Course Goals: To cover the basic principles and practical examples of heat and mass transfer Learning Outcomes (subject to change) Week 1

! To know the units for different variables: flux, rate, thermal conductivity, etc. ! To distinguish thermal energy (and heat transfer) from other forms of energy (and energy

transfer) ! To perform general energy balance and surface energy balance ! To explain the role of each term of the general energy balance equation ! To recognize the different forms of heat transfer ! To define the different modes of heat transfer ! To memorize the different heat rate equations for

o conduction (Fourier’s Law of heat diffusion) 𝒒!= −𝑘∆𝑇

convection (Newton’s Law of cooling) q/A=hΔT o radiation (Stefan Boltzmann law of radiation) q/A=σT4

! To remember the orders of magnitude of thermal conductivities ! To apply the heat transfer equations and solve simple heat transfer problems

o to calculate a heat flux by convection using Newton rate equation if an appropriate temperature difference and heat transfer coefficient, h, are given

o to calculate a temperature if a convective heat flux, one temperature, and a heat transfer coefficient, h, are given

o to calculate the heat flux of a black body using the Stefan-Boltzmann equation if a temperature is given

o to calculate the temperature of a black body using the Stefan-Boltzmann equation if a radiative heat flux is given

! To be able to transcript mathematically what “steady state conditions” mean ! To remember and use the methodology presented in tutorial to solve a heat transfer

problem, which involves the sections: Known, Find, Schematic, Assumptions, Properties, Analysis, Comments

Week 2

! To be able to simplify heat transfer problems by identifying and implementing appropriate approximations

! To obtain the differential equation of heat conduction in Cartesian coordinate system ! To be able to simplify the differential equation of heat conduction for one dimensional

steady-state cases in all coordinate systems ! To be able to integrate the heat equation and find the heat flux ! To be able to explain each term of the conduction heat transfer equation in all coordinate

systems ! To solve one dimensional steady-state heat conduction problems and obtain the

temperature distributions and the flux within a medium (e.g., T=f(x)) ! To calculate the flux at any point in the object using Fourier rate law using the temperature

profile, e.g., first use T=f(x) to find dT/dx, second calculate q/A=-kdT/dx ! To know the units of thermal resistance and thermal conductivities ! To be able to set, identify, and transcribe mathematically boundary and initial conditions

for heat transfer problems. For example: 1. Constant temperature T=const, 2. Constant flux, with a special case of dT/dx=0 for perfect insulation or for symmetry lines and 3. Flux matching.

! To understand what negative or positive value of the heat flux means with respect to a given coordinate system (e.g. positive, heat flows in the same direction as axis orientation).

! To be able to simplify the generalized energy balance equation in the case with internal energy generation, to come up with a correct governing equation for the temperature profile in a differential form. For example, go from:

k∇!𝑇 + 𝑞 = 𝜌𝐶!  𝜕𝑇/𝜕𝑡  to  

!!!!!!

+ 𝑞/𝑘 = 0  

Week 3

! To understand the concept of thermal resistance and its limitations ! To be able to draw an equivalent electrical circuit that will show resistances to heat

transfer, and analyze heat transfer problems using the concept of thermal resistances. Resistances can be connected in series, in parallel or in combination. Resistances could be due to convection, conduction, or radiation.

! To solve steady state 1-D heat conduction problems that involve multiple layers of material in all coordinate systems

o To be able to find heat flux for the entire circuit as well as fluxes in the pieces of circuit, if temperature differences are given, as well as the dimensions and thermal conductivities k for the solid objects and heat transfer coefficients, h, for fluids in the equivalent electrical circuit

o To be able to find temperatures at fixed points in the circuit around the resistances using previously determined heat fluxes (e.g. if the overall temperature difference is given, each student should be able to find overall heat flux, fluxes through each resistor and temperature differences around each resistor).

o To be able to find overall temperature difference in the equivalent circuit and temperature difference around each resistor, if a flux is given, as well as dimensions, k, and h

! To memorize the expressions for resistances in Cartesian coordinates: R=1/hA, R=L/kA ! To memorize the expressions for resistances in cylindrical coordinates: R=1/hA,

R=ln(ro/ri)/2πkL. Note that A varies with position ! To memorize the expressions for the sum of resistances in series (Rtot=R1+R2+…) and in

parallel (1/Rtot=1/R1+ 1/R2 etc…) ! To memorize and be able to calculate the overall heat transfer coefficient U=1/(Rth, tot.A)

Weeks 4, 5, 6

! To analyze finned surfaces, and assess how fins can enhance heat transfer Un-steady state Un-steady state, Bi≤0.1

! To memorize the formula for Biot number: Bi=h(V/A)/k or Bi=hLc/k ! To determine if the lumped parameter approach applies in a given problem based on the

value of Bi (Bi≤0.1) ! To assess when the spatial variation of temperature is negligible, and temperature varies nearly uniformly with time, making the simplified lumped system analysis applicable.

! To find the time needed to reach a given temperature in a solid object if initial and final temperatures are given, as well as the properties of the solid object (dimensions, ρ, Cp), the temperature of the surrounding fluid, and the convective heat transfer coefficient in the fluid

! To memorize solved energy balance for lumped parameter approach

𝒍𝒏𝑻− 𝑻!𝑻𝒊 − 𝑻!

= −𝒉𝑨𝝆𝑽𝑪𝒑 𝒕

! To calculate the temperature of the solid object reached in a period of time if initial temperature is given, as well as the properties of the solid object (dimensions, ρ, Cp), the temperature of the surrounding fluid, and the convective heat transfer coefficient in the fluid

! To calculate the total energy transfer during a given time for cases where the lumped capacitance method can be used

! To memorize the formula for the Fourier number: Fo = αt/L2 ! General lumped capacitance analysis:

o To be able to conduct and simplify the energy balance for the general case where temperature varies as a function of time only, and where a constant surface heat flux is applied to one side of a solid, a convection surface condition applied on another side, and radiation is present. To obtain an expression for the differential equation.

o To be able to integrate the differential equation of the general lumped capacitance analysis in the case of negligible radiation, by introducing the transformation θ’ = θ – b/a

Un-steady state, high Biot number ! To understand the different governing equations for the temperature profile in the

differential form and with different boundary conditions ! To understand and be able to write down the boundary and initial conditions for the

temperature T, and their equivalence for the dimensionless temperature θ∗ in the case of a plane wall of thickness 2L.

! To know when the approximate solutions (first term of the infinite series) of the heat equation (!

!!∗

!"∗!= !!∗

!"#) can be used (for Fo < 0.2) for a plane wall and a radial system with

convection ! To understand the approximate solutions of the heat equation (!

!!∗

!!∗!= !!∗

!"#) for a plane wall

and a radial system with convection, and be able to use given tables to find the values for C1, ζ, J1(x), and Jo(x)

! After finding the values in appropriate tables, to be able to calculate a temperature after a given time, a time to reach a certain temperature, and the amount of energy transferred in a given time.

! Using time-temperature charts, each student should be able to find: o The temperature at any point within a given object after a period of time, if all

material properties, the initial temperature of the solid object and the temperature of the fluid around the solid object are given.

o The time it takes to reach certain temperature at a given position in the object, if all material properties, the initial temperature of the solid object and the temperature of the fluid around the solid object are given.

Weeks 8 - 10: Introduction to Convection & External Forced Convection ! To understand the physical mechanism of convection, its classification (natural vs. forced,

internal vs. external), and the differences with conduction ! To visualize the development of velocity and thermal boundary layers during flow over

surfaces ! To be able to calculate the thickness of the velocity (p. 379) and thermal (p. 380) boundary

layers based on either the definition or Blasius’ solutions of differential equation (p. 439 Equation 7.19; p. 441 Eq. 7.24)

! To memorize the definition and gain an understanding of the wall shear stress and friction coefficient: 𝜏! = 𝜇 !"

!" !!! and 𝜏! = 𝐶!𝜌

!!!

!

! To memorize the definition and gain an understanding of the heat transfer coefficient:

ℎ =−𝑘!"#$%×

𝜕𝑇𝜕𝑦 !!!

𝑇! − 𝑇!

! To use and simplify the generalized heat transfer equation with convection, to find a temperature profile using appropriate boundary conditions

𝜌𝐶!𝜕𝑇𝜕𝑡 + 𝒗 ∙ ∇𝑇 = k∇!𝑇 + 𝑞  

! To memorize and understand the simplified boundary layer equations and for continuity, momentum, and energy (p. 395)

! To understand the meaning of the different terms of the continuity, momentum, and energy equations

! To be able to write and gain a working understanding of the normalized forms of the boundary layer continuity, momentum, and energy equations (Table. 6.1, p. 399)

! To memorize, understand the meaning, and gain a working knowledge of the dimensionless Reynolds, Prandtl, Stanton, Peclet, and Nusselt numbers.

! To use analogies between momentum and heat transfer, and determine heat transfer coefficient from knowledge of friction coefficient

! To memorize the Reynolds analogy (Pr = 1) and understand the connection between momentum and energy transfer: Nu,x = Cf,x ReL/2

! To memorize the Chilton-Colburn analogy (0.6 < Pr < 60) and understand the connection between momentum and energy transfer: Nu,x Pr -1/3= Cf,x ReL/2

! To distinguish between laminar and turbulent flows, know the appropriate values for the critical Reynolds number in external forced convection, and to understand the mechanisms of momentum and heat transfer in turbulent flow

! To be able to conduct energy balances that involves a term related to heat energy (H=enthalpy) transport by a moving mass (𝑚=mass flowrate=density*surface*Velocity of moving mass) = H*𝑚 in order to derive the differential equations that govern convection, and solve these equations for some simple cases such as laminar flow over a flat plate (e.g., Problem Book # 7.28 = Pb#6 Handout Week 9)

! To develop an intuitive understanding of friction drag and velocity and thermal boundary layers, and evaluate the local & average friction and convection coefficients in external flow

! To be able to find h (note that h is contained within Nu or St), in the case of forced convection, external laminar flow, flat plate, using an exact analysis of the boundary layer for plug flow over a flat plate, and to memorize the following expression, and understand where it comes from: NuL=0.664ReL

1/2Pr 1/3 ! To be able to find h for different geometries and flow regimes, and calculate the rate of

heat transfer ! To calculate and understand when to use the film temperature Tf = (Ts + T∞)/2

Week 11: Internal Forced Convection

! To distinguish between laminar and turbulent flows and know the appropriate values for the critical Reynolds number in internal forced convection

! To be able to calculate the hydrodynamic entry length for laminar and turbulent flow, and know when it is reasonable to assume that the flow is fully developed.

! To know the definition and requirement for fully developed hydrodynamic and thermal conditions (e.g., Equation 8.9 p. 520 and Equation 8.28, p. 526)) and consequences, i.e., constant local convection coefficient (Equation 8.29 p. 526)

! To memorize the definition of the mean velocity (𝑚 = 𝜌𝑢!𝐴!) and mass flow rate (𝑚 = 𝜌𝑢 𝑟, 𝑥 𝑑𝐴!!!

), and therefore get an expression for um.

! To memorize the definition of the true rate of thermal energy (enthalpy) advection integrated over the cross-section and the mean temperature (𝑚𝐶!𝑇!) (𝑚𝐶!𝑇! =

𝜌𝑢 𝑟, 𝑥 𝐶!𝑇(𝑟, 𝑥)𝑑𝐴!!!), and therefore get an expression for Tm.

! To memorize the expression for the Newton’s law of cooling q”s = h(Ts-Tm) for internal flow and see the similarity (q”s = hΔT) and differences (while T∞ is constant in the flow direction, Tm is not) with that for external flow: q”s = h(Ts-T∞)

! To be able to find the friction coefficient by using the right expression or reading on the Moody diagram (p. 522-523), and to calculate the pressure drop and power requirement

! To be able to conduct energy balances involving thermal energy advection (= convection) and therefore get an expression for the axial (with x) variation of the mean temperature Tm depending on the surface boundary conditions (p. 529 – 535)

! To understand the origin of the log mean temperature difference and remember its expression

! To recall that all correlations for Nu and criteria to determine laminar vs. turbulent flow are associated with some error.

! To be able to find h (note that h is contained within Nu or St), for a variety of conditions in internal forced convection, in the fully developed and entry regions, using the appropriate correlation and reading charts.

! Each student should know the definition of the Graetz number and find Nu number as a function of Gz number.

! To understand how to determine the hydraulic diameter for non-circular pipe and concentric tubes.

Week 12: Heat Exchangers ! To recognize numerous types of heat exchangers and classify them ! To determine the overall heat transfer coefficient for a heat exchanger ! To memorize the expression for the overall fin efficiency (Equation 11.3 p. 709, and 11.4

p. 710, the expression for m = sqrt(2h/kt) will be given) ! To memorize the form of the steady state flow energy equation describing the amount of

energy given out or received per unit of time by any stream in the heat exchanger: q=ṁCp (Tin-Tout)

! To determine the amount of heat transferred between the two streams in a heat exchanger, using the following methods, provided overall heat transfer coefficient, U, and area available for the heat exchange, A, are known:

o To memorize and use the Log mean temperature difference (LMTD) for parallel and counter-flow operation

q=U A ΔTlm (Each student needs to know how to use the formula for ΔTlm, in parallel or counter-flow.)

o To memorize and use the equation involving the correction factor F for shell-and-tube or cross-flow heat exchangers

q=U A F ΔTlm (Charts will be given, each student needs to know how to use the charts to get F, see pdf document posted on blackboard)

o To memorize and use the Number of Transfer Units (NTU) method: NTU=UA/Cmin

Cmin=ṁCp, for Cp: pick smaller value of two. q=εCmin(Thot in - Tcold in)

where ε is the heat transfer effectiveness, and is a function of NTU and Cmin. It can be found using equations for parallel or counter-flow, as well as by using charts for shell-tube and cross-flow heat exchangers. Equations (e.g., Tables 11.3 And 11.4)

and charts (e.g., Fig. 11.10 to 11.15) will be provided, each student is responsible for knowing how to use them.

! To find the required heat exchange area, A, using the equation above, and provided q, U, and the appropriate temperatures are known.

! To understand that the best heat exchanger is the one that uses the smallest surface area to accomplish a given heat transfer rate

! To understand that fouling can lead to decrease in the overall heat transfer coefficient U. Fouling resistance Rf can be included in the overall heat transfer coefficient using electrical analogy.

! To remember how to calculate thermal resistances for convection and conduction for tubes. Weeks 13 & 14: Mass Transfer

! To understand the physical mechanism of mass transfer by convection (or “advection”, due to bulk fluid motion) and diffusion (due to a concentration difference).

! To recognize the analogies between heat and mass transfer ! To memorize and understand the mass conversation equation ! To memorize, understand, use, and simplify the mass diffusion equation ! To know that when diffusion is the only mode of mass transfer, the mass flux can be

calculated according to the Fick’s law of diffusion, applicable for a dilute, binary mixture, of constant density, and with a constant diffusion coefficient:

𝐽 = −𝐷!"𝛁𝐶 ! To solve

o 1-D steady state diffusion o 1-D unsteady state diffusion, for (i) constant diffusion coefficient, (ii) no

homogeneous reaction, (iii) initial constant concentration, and (iv) a semi-infinite medium, using the solution of mass diffusion equation below

𝐶!"  –𝐶!𝐶!"  –𝐶!"

= erf𝑥

2 𝐷!"    𝑡

! To be aware that discontinuities in the concentrations in the two phases of an interface can appear, e.g., between liquid and gas, liquid and solid. This depends on the solubility of a species in the different phases, which is describes by partitioning coefficients K (for liquid – solid), using the Henry’s law coefficient Hcc (for liquid – gas), the Raoult’s law, and the definition of solubility. To memorize and be able to use these interface relations.

! To know that when convection is the dominant mode of mass transfer, the mass rate can be calculated using the following equation: NA=u∞CAAs

! To calculate the mass rate of species A by convection if the mass transfer coefficient hm is known: NA=hmAs(CA,s - CA,∞) Where hm can be obtained using a similar reasoning as for h: exact analysis of the boundary layer, analogies between momentum, energy and mass transfer or correlations

Project specific Learning Outcomes Once the project will be completed, the students

! will be able to conduct structured team work ! will be able to apply the principles of professional writing while producing an engineering

report ! will be able to document the process and choices made in designing the equipment ! will be able to state appropriate assumptions, propose mathematical models, and select

equations and correlations accordingly ! will be able to design an equipment or system for a realistic heat transfer problem ! will be able to assess the effects of changes in operating conditions on the performance of

the proposed design and discuss on solution alternatives ! will be able to work and keep regular deadlines and learn how to organize work