University:
Massachusetts Institute of Technology
Course:
2.141 | Modeling and Simulation of Dynamic Systems
Academic year:

2022
Views:

154

Pages:

21
Author:

Ernest Stein

HEAT TRANSFER AND THE SECOND LAW Thus far we’ve used the first law of thermodynamics: Energy is conserved. Where does the second law come in? One way is when heat flows. Heat flows in response to a temperature gradient. If two points are in thermal contact and at different temperatures, T1 and T2 then energy is transferred between the two in the form of heat, Q. The rate of heat flow from point 1 to point 2 depends on the two temperatures. Q̇ = f(T1,T2) If heat flows from hot to cold, (the standard convention) this function must be such that Q̇ > 0 iff T1 > T2 Q̇ < 0 iff T1 < T2 Q̇ = 0 iff T1 = T2 In other words, the relation must be restricted to 1st and 3rd quadrants of the Q̇ vs. T1 – T2 plane. . Q T1- T2 NOTE IN PASSING: It is not necessary for heat flow to be a function of temperature difference alone. —see example later. HEAT FLOW GENERATES ENTROPY. FROM THE DEFINITION OF ENTROPY: dQ = TdS Therefore Q̇ = TṠ IDEALIZE THE HEAT TRANSFER PROCESS: Assume no heat energy is stored between points 1 and 2. Therefore Q̇ = T1Ṡ 1 = T2Ṡ 2 NET RATE OF ENTROPY PRODUCTION: entropy flow rate out minus entropy flow rate in. Ṡ 2 - Ṡ 1 = Q̇ /T2 - Q̇ /T1 = Q̇(T1 - T2) /T1T2 Absolute temperatures are never negative (by definition). The product of heat flow rate and temperature difference is never negative. —due to the restrictions on the relation between heat flow rate and temperature (Q̇ > 0 when T1 > T2 etc.) THEREFORE THE NET RATE OF ENTROPY PRODUCTION IS NEVER NEGATIVE. Ṡ 2 - Ṡ 1 ≥ 0. FURTHERMORE... In a heat transfer process, zero entropy production requires zero heat flow. This requires either a perfect thermal insulator (Q̇ = 0) or a zero temperature gradient (T1 = T2). These two constraints on the heat transfer process —non-negative entropy production —zero entropy production iff zero heat flow are consequences of the second law of thermodynamics. NOTE IN PASSING: Many processes other than heat transfer also generate entropy —see example later. EXAMPLE: CONDUCTIVE HEAT TRANSFER THERMAL CONDUCTION: flow of heat from one body to another through direct contact. A COMMON MODEL OF CONDUCTIVE HEAT TRANSFER: heat flow rate is proportional to temperature difference (sometimes called Fourier’s Law) For an insulated rod conducting through it ends kA Q̇ = l (T1 - T2) where A is area l is length k is thermal conductivity WHAT TYPE OF NETWORK ELEMENT DESCRIBES THIS PHENOMENON? THE MODEL RELATES POWER FLOW TO EFFORT. Rewrite this relation in terms of efforts, flows, displacements, etc. ASSUMING AN IDEAL HEAT TRANSFER PROCESS (no heat energy is stored between points 1 and 2) Q̇ = T1Ṡ 1 = T2Ṡ 2 From the model above (Fourier’s Law) we get two constitutive equations. kA Ṡ 1 = l (T1 - T2) /T1 kA Ṡ 2 = l (T1 - T2) /T2 THE TWO CONSTITUTIVE EQUATIONS ARE COUPLED Ṡ 1 = Ṡ 1(T1,T2) Ṡ 2 = Ṡ 2(T1,T2) This is a two-port element. THE EQUATIONS RELATE EFFORTS (TEMPERATURES) AND FLOWS (ENTROPY FLOW RATES) — this is a two-port resistor. The bond graph symbol is T1 dS1/dt R T2 dS2/dt NOTE: Absolute temperatures are never negative, but either (or both) of the two entropy flow rates Ṡ 1 and Ṡ 2 may be negative—e.g., when T2 > T1 However, the net entropy production rate is never negative. kA Ṡ 2 - Ṡ 1 = l (T1 - T2)(1/T2 – 1/T1) kA Ṡ 2 - Ṡ 1 = l (T1 - T2) 2/T1T2 Ṡ 2 - Ṡ 1 ε 0 —This is as required by the second law. —The constitutive equations automatically satisfy the second law. THIS SIMPLE TWO-PORT RESISTOR —adds entropy production behavior to our models —ensures the second law is satisfied EXAMPLE: RADIATIVE HEAT TRANSFER THERMAL RADIATION: the flow of heat from one body to another without direct contact. A SIMPLE MODEL OF RADIATIVE HEAT TRANSFER heat flow rate is proportional to the difference of the fourth powers of the temperatures. (Stefan-Boltzmann Law) Q̇ = σ(T14 - T24) σ: radiative heat transfer coefficient. T1 and T2: absolute temperatures. This model satisfies the restrictions on the relation between heat flow rate and temperature difference. Q̇ = 0 iff T1 = T2 Q̇ > 0 iff T1 > T2 Q̇ < 0 iff T1 < T2 NOTE: This is a case in which the heat flow rate can not be expressed as a function of the temperature difference alone —as advertised above. TO FIND CONSTITUTIVE EQUATIONS FOR THIS PROCESS, PROCEED AS BEFORE: Assume an ideal heat transfer process (no heat energy stored between points 1 and 2) Q̇ = T1Ṡ 1 = T2Ṡ 2 Again we get two constitutive equations Ṡ 1 = σ(T14 - T24)/T1 Ṡ 2 = σ(T14 - T24)/T2 As before, these two constitutive equations are coupled and describe a twoport resistor T1 dS1/dt R T2 dS2/dt As before, either or both individual entropy flow rates Ṡ 1 and Ṡ 2 may be negative—e.g., when T2 > T1 However, the constitutive equations guarantee non-negative net entropy production. Net entropy production rate Ṡ 2 - Ṡ 1 = σ(T14 - T24)(1/T2 – 1/T1) Expand this expression Ṡ 2 - Ṡ 1 = σ(T12 + T22)(T1 + T2)(T1 – T2)(1/T2 – 1/T1) = σ(T12 + T22)(T1 + T2)(T1 – T2)2/T1T2 ε 0 Because absolute temperatures are never negative, net entropy production rate is never negative. Ṡ 2 - Ṡ 1 ε 0 —as required by the second law. CAUSALITY AND THE SECOND LAW CAUSAL ANALYSIS — INTRODUCED EARLIER — IS AN IMPORTANT ASPECT OF PHYSICAL SYSTEM MODELING. To model physical processes, we represent relations between quantities as operations on variables. The choice of which variables may be used as input and which as output is not arbitrary. The definition of an element may prohibit some choices. e.g., its constitutive equation may have no inverse. There may be a “natural” preference for some choices. e.g., the integral causal form is preferable for an energy-storage element. THERE IS A RELATION BETWEEN CAUSAL FORMS AND THE SECOND LAW OF THERMODYNAMICS. Regarded as assignment operators, the heat transfer constitutive equations have temperature inputs and entropy flow rate outputs on both ports. T1 dS1/dt R T2 dS2/dt —”conductance” causality (effort in, flow out) (rather than “resistance” causality) THIS CAUSAL FORM SATISFIES THE SECOND LAW DUE TO (a) the constitutive equations and (b) the sign-definiteness of absolute temperature. THE ALTERNATIVE CAUSAL FORMS DO NOT HAVE THIS OBVIOUS RELATION TO THE SECOND LAW. For example: Re-consider the model of conductive heat transfer change one port to a “resistance” causal form (entropy flow rate in, temperature out) T1 dS1/dt R T2 dS2/dt A little algebra shows the corresponding constitutive equations to be: Ṡ 1 = T2 = ⎛kA⎞ ⎜ ⎟ ⎜ ⎟ ⎝ l ⎠ ⎛kA⎞ ⎟ ⎜ ⎜ ⎟ ⎝ l ⎠ ⎛ Ṡ2 ⎞ ⎜Ṡ + kA⎟ ⎝2 l⎠ T1 kA Ṡ2 + l The first equation suggests that entropy flow Ṡ 1 is independent of temperature. That seems odd—we usually think of entropy (and heat) as flowing in response to a temperature gradient. The second equation suggests that temperature T2 may be infinite if Ṡ 2 = –kA/l or negative if Ṡ 2 < –kA/l Those values of temperature would have little physical meaning. FURTHERMORE... ACCORDING TO THESE EQUATIONS, THE NET ENTROPY PRODUCTION RATE IS Ṡ22 Ṡ 2 - Ṡ 1 = kA Ṡ2 + l This equation suggests that net entropy production may be negative if Ṡ 2 < – kA/l — that would violate the second law. These problems might be solved by restricting the value of the input entropy flow rate. Ṡ 2 > –kA/l THAT WOULD BE UNSATISFACTORY. 1. Even with that constraint, this form still suggests that entropy flow Ṡ 1 is independent of temperature — that does not facilitate insight. 2. We don't know a-priori which other elements in a system generate the input entropy flow — it is not clear where to apply the constraint. 3. The constraint on entropy flow rates is due to the physical properties of heat transfer. If our model is to reflect our understanding of the physical process, the constraint should be associated with the element used to model heat transfer, — not with the rest of the system. In contrast, the “conductance” causal form (temperature inputs, entropy flow rate outputs) satisifies the constraints of the second law through the constitutive equations of the resistive multiport element itself. In fact, this is the only causal assignment which will guarantee that the resistive multiport will satisfy the second law by virtue of its own constitutive equations. Thus there is a “natural” preference for the conductance causal form for an entropy-producing multiport resistor. (Analogous to the preferred integral causal form for an energy storage element.)

Heat Transfer and The Second Law

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